Advances in Topology and Their Interdisciplinary Applications 9819901502, 9789819901500

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Industrial and Applied Mathematics

Santanu Acharjee   Editor

Advances in Topology and Their Interdisciplinary Applications

Industrial and Applied Mathematics Editors-in-Chief G. D. Veerappa Gowda, TIFR Centre For Applicable Mathematics, Bengaluru, Karnataka, India S. Kesavan, Institute of Mathematical Sciences, Chennai, Tamil Nadu, India Fahima Nekka, Universite de Montreal, Montréal, QC, Canada Editorial Board Akhtar A. Khan, Rochester Institute of Technology, Rochester, USA Govindan Rangarajan, Indian Institute of Science, Bengaluru, India K. Balachandran, Bharathiar University, Coimbatore, Tamil Nadu, India K. R. Sreenivasan, NYU Tandon School of Engineering, Brooklyn, USA Martin Brokate, Technical University, Munich, Germany M. Zuhair Nashed, University of Central Florida, Orlando, USA N. K. Gupta, Indian Institute of Technology Delhi, New Delhi, India Noore Zahra, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia Pammy Manchanda, Guru Nanak Dev University, Amritsar, India René Pierre Lozi, University Côte d’Azur, Nice, France Zafer Aslan, ˙Istanbul Aydın University, ˙Istanbul, Türkiye

The Industrial and Applied Mathematics series publishes high-quality researchlevel monographs, lecture notes, textbooks, contributed volumes, focusing on areas where mathematics is used in a fundamental way, such as industrial mathematics, bio-mathematics, financial mathematics, applied statistics, operations research and computer science.

Santanu Acharjee Editor

Advances in Topology and Their Interdisciplinary Applications

Editor Santanu Acharjee Department of Mathematics Gauhati University Guwahati, Assam, India

ISSN 2364-6837 ISSN 2364-6845 (electronic) Industrial and Applied Mathematics ISBN 978-981-99-0150-0 ISBN 978-981-99-0151-7 (eBook) https://doi.org/10.1007/978-981-99-0151-7 Mathematics Subject Classification: 54-XX, 22-XX, 26Axx, 46Axx, 91A44 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Mathematics deals with abstract ideas, geometrical objects, physical phenomena, etc. But when one thinks of rubber sheet geometry, almost all the mathematical areas other than “topology” fail to explain this geometry and related properties. Topology is used in almost all areas of science and social science. The presence of topological connections in several areas of biology, chemistry, computer science, economics, neuroscience, physics, etc., has been opening ways for topology in other areas since the middle of the twentieth century. At present, the progress of topology is too fast to understand the cutting-edge research trends. It is well known that the twenty-first century is the century of big data analytics, block-chain technology and artificial intelligence. Thus, the presence of topology can be found also in these areas. In short, topology is everywhere. So, it is the need of the hour to provide a suitable platform to the experts of different subjects for the exchange of ideas related to their respective fields from the perspective of topology. Hence, this book fulfills the need. This book tries to cover the gaps between recent theoretical developments and interdisciplinary applications of topology. It contains fourteen chapters and each chapter opens a new horizon of research in topology for the researchers of the present and next generations. As new trends emerge, chapters related to interdisciplinary subjects, such as economics, theoretical chemistry, cryptography, pattern recognition, granular computing, etc., are included along with chapters on mathematical topics like function spaces, relator space, preorder, quasi-uniformities, bitopological dynamical systems, b-metric spaces and related fixed point theory, etc. Chapters in the book are independent, self-sufficient and written by well-known experts in the related fields. Each chapter starts with a self-sufficient introduction and then proceeds to its advanced level. These significant features also make this book useful for graduate students, researchers and experts in various interdisciplinary fields along with mathematics. In Chap. 1, the authors prove that spaces of minimal usco and minimal cusco maps from a locally compact space to a Fréchet space, equipped with the topology

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of uniform convergence on compacta are isomorphic as topological vector spaces. Moreover, if the domain space is also hemicompact, both spaces are Fréchet. Chapter 2 studies contra-continuity properties in relator space. Contra-continuity and some of the related ideas are available in general topology, but they were not studied extensively in the relator space. In this chapter, several new results in relator space are studied with contra-continuity. A topological space (X, τ ) satisfies the continuous representation property (CRP) if every continuous total preorder defined on X can be represented by a continuous order-preserving real-valued function. In Chap. 3, the relevance of this property is discussed in the context of economics and social sciences. Certain characterizations of CRP are presented in terms of other familiar topological properties. In addition, two extensions of CRP, one regarding the semi-continuous case and the other involving an algebraic environment, are also discussed. Chapter 4 aims to present remarks and questions on quasi-uniformities on function spaces F(X, Y), that are generated by atoms. Although this type of quasi-uniformities can be easily defined, the corresponding topologies do not behave well with respect to the well-known exponential laws. These quasi-uniformities are not necessarily be atoms, thus author discusses the problem of defining a correspondence between the atoms of the lattice of quasi-uniformities and those atoms that are in the lattice of all quasi-uniformities on Y, in the function space F(X, Y). In Chap. 5, inclusion–exclusion formula and interval arithmetic are used to obtain interval estimations of cardinal numbers of certain basic sets in the finite topological spaces satisfying Kolmogorov’s T 0 separation axiom. The elaboration of preference relations and their representations take back their source in early economic theory. Classical representations of preferences theorems rely on Debreu–Eilenberg’s theorems through abstract mathematics based on topological properties. In Chap. 6, another approach is adopted starting from metric spaces instead. The author obtains representation theorems of preference relations with bivariate functions. This allows economists to handle intransitivities and incomparabilities of the preference relation and also continuity conditions of various strengths. In Chap. 7, entropy in non-weakly pairwise compact bitopological dynamical systems is introduced as a measure of complexity and several new results related to entropy are studied. Also, possible connections between the neural activities of the human brain and the entropy of a pairwise continuous map in NWPC bitopological dynamical systems are discussed. In Chap. 8, the authors discuss several variants of variational inequality problems that exist in the literature. They also discuss solutions to the vector variational inequality problem and generalized vector variational inequality problem in a more general framework by using a topological approach. In Chap. 9, the authors introduce the interior and the closure operators with respect to an ideal defined on an approximation space generating an ideal approximation space. Separation axioms and connectedness in approximation spaces and in ideal

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approximation spaces are defined and some properties are studied. Also, grill separation axioms and grill approximation connectedness with respect to a given grill are defined and various properties are studied. Chapter 10 searches the equivalent definition of Kuratowski’s local function in terms of a filter and studies some properties between big set, small set, homeomorphism, etc. As an application, a secret information-sharing scheme is developed in topological cryptography. In Chap. 11, Fisher-type set-valued mappings in a b-metric space are considered and some fixed point results considering Pompeiu–Hausdorff metric with respect to the b-metric are derived. A stability result for the fixed point sets is obtained by using some topological characteristics of such mappings. An application is also shown for the existence of a solution to an integral inclusion of the Fredholm type. Granular computing is an area of theoretical computer science that is highly useful in nature. Recently, several new directions of granular computing have been explored with crisp set, fuzzy set, rough set and related hybrid structures. In Chap. 12, the authors try to explore various ideas related to granularity from the perspectives of general topology, binary relations and crisp sets only. At last, some feasible ideas from biology and microscopy are discussed, which may inspire granular computing to have new theories based on crisp sets and reality of nature. A topological index is a numeric quantity that characterizes the whole structure of a chemical graph. In Chap. 13, Randi´c index and its generalized forms, first Zagreb index, augmented Zagreb index, harmonic index, hyper Zagreb index, atom-bond connectivity index, geometric-arithmetic index, forgotten index are computed for zeolite material [4, 2] and zeolite material [4, n], where n ≥ 3. Moreover, the fourth version of the atom–bond connectivity index and the fifth version of the geometric– arithmetic index of zeolite material [4, n], where n ≥ 5, are also calculated. In Chap. 14, the authors introduce the concept of q-rung orthopair fuzzy point and propose a Dice similarity measure and a distance measure between q-rung fuzzy sets by using the concept of Choquet integral. Some applications of pattern recognition by using q-rung orthopair fuzzy points and the Dice similarity measure are done from the perspective of fuzzy topology. At the end, I shall be fortunate enough if this book may find its suitable place on the tables of researchers who are interested in learning and working on recent trends in topology. Finally, it is now time to acknowledge the various supports that I received while editing this book. I would like to thank Gauhati University for providing me various supports during the tenure of editing this book. I am also thankful to my friends and well-wishers from both academia and non-academia for their various encouragements. Mr. Shamim Ahmad, Executive Editor of Springer, deserves special thanks for showing his prompt interest in my proposal to edit a book on topology. I am also thankful to the rest of the members of Springer for their various supports.

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My acknowledgments will remain incomplete if I fail to thank my mother and my wife for their various encouragements and moral supports. At the extreme point of acknowledgment, I am thankful to the Almighty for choosing me to become a topologist. Long live topology! Guwahati, India

Santanu Acharjee

Contents

1

Spaces of Minimal Usco and Minimal Cusco Maps as Fréchet Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L’ubica Holá, Dušan Holý, and Branislav Novotný

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Contra Continuity Properties of Relations in Relator Spaces . . . . . . Árpád Száz

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The Continuous Representation Property in Utility Theory . . . . . . . Juan C. Candeal

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On Quasi-uniformities, Function Spaces and Atoms: Remarks and Some Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angelika Kontolatou, Kyriakos Papadopoulos, and John Stabakis

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Some Cardinal Estimations via the Inclusion-Exclusion Principle in Finite T0 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 103 James F. Peters and Irakli J. Dochviri

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Representations of Preference Relations with Preutility Functions on Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Yann Rébillé

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Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Santanu Acharjee, Kabindra Goswami, and Hemanta Kumar Sarmah

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Topological Approaches for Vector Variational Inequality Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Ankit Gupta, Satish Kumar, and Pankaj Kumar Garg

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Ideals and Grills Associated with a Rough Set . . . . . . . . . . . . . . . . . . . . 167 I. Ibedou, S. E. Abbas, and S. Jafari

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Contents

10 Filter Versus Ideal on Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 183 Jiarul Hoque, Shyamapada Modak, and Santanu Acharjee 11 Fisher Type Set-valued Mappings in b-metric Spaces and an Application to Integral Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . 197 Nilakshi Goswami and Nehjamang Haokip 12 Topological Aspects of Granular Computing . . . . . . . . . . . . . . . . . . . . . 217 Santanu Acharjee, Amlanjyoti Oza, and Upashana Gogoi 13 On Topological Index of Naturally Occurring Zeolite Material [4, n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Saitanya K. Bharadwaj and Santanu Acharjee 14 q-Rung Orthopair Fuzzy Points and Applications to q-Rung Orthopair Fuzzy Topological Spaces and Pattern Recognition . . . . . 245 Ezgi Türkarslan, Mehmet Ünver, Murat Olgun, and Seyhmus ¸ Yardımcı

About the Editor

Santanu Acharjee is an Assistant Professor at the Department of Mathematics, Gauhati University, Guwahati, Assam, India. He did his M.Sc. and Ph.D. in Mathematics from Gauhati University. His research areas are topology, soft computing, artificial intelligence, mathematical social science, mathematical economics, social networks, big data analytics, human trafficking and antiterrorism research. He has published more than 30 research papers and 3 book chapters. Moreover, he has collaborated with several eminent researchers and visiting researchers of several international institutes: the Institute for Advanced Study, USA; the University of Oxford, UK; Creighton University, USA; the University of Auckland, New Zealand; Kuwait University; the University of California, Riverside, USA; the Russian Academy of Science; and the University of Debrecen, Hungary, etc. Dr. Acharjee was invited as a Visiting Researcher by the Fields Institute, Canada. He was awarded a travel grant by the National Board of Higher mathematics (NBHM). He jointly introduced a new area of mathematical research named “bitopological dynamical system” in the year 2020. He is a member of the American Mathematical Society (USA) and the International Association of Engineers (Hong Kong). He is also a life member of the Indian Science Congress Association (India). Dr. Acharjee is a reviewer for Mathematical Reviews (AMS), zbMATH Open (Germany), several journals of American Psychological Association as well as 38 other international journals of mathematics and interdisciplinary fields. He has edited special issues for some reputed journals as a guest editor. He jointly edited the book entitled Advances in Mathematical Analysis and its Applications. He has delivered more than 10 invited talks at various national and international conferences.

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Chapter 1

Spaces of Minimal Usco and Minimal Cusco Maps as Fréchet Topological Vector Spaces L’ubica Holá, Dušan Holý, and Branislav Novotný

Abstract Minimal USCO/CUSCO maps are very important in functional analysis, optimization, the study of differentiability of Lipschitz functions, etc. Therefore it is important to know their topological properties. One of the most important topology on function spaces is the topology of uniform convergence on compacta. We will prove that spaces of minimal usco and minimal cusco maps from a locally compact space to a Fréchet space, equipped with the topology of uniform convergence on compacta are isomorphic as topological vector spaces. Moreover, if the domain space is also hemicompact, both spaces are Fréchet. Keywords Minimal USCO map · Minimal CUSCO map · Topology of uniform convergence on compacta · Fréchet topological vector space · Hemicompact space

1.1 Introduction Minimal usco maps have probably first appeared in complex analysis in the second half of the 19th century as bounded holomorphic functions and their “cluster sets”, see e.g. [10]. Minimal cusco maps are very important in functional analysis, where the differentiability of Lipschitz functions is deduced by their Clarke subdifferentials being minimal cuscos, see [5].

L. Holá (B) · B. Novotný Slovak Academy of Sciences, Institute of Mathematics, Štefánikova 49, Bratislava, Slovakia e-mail: [email protected] B. Novotný e-mail: [email protected] D. Holý Department of Mathematics and Computer Science, Faculty of Education, Trnava University, Priemyselná 4, 918 43 Trnava, Slovakia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_1

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Here is a short overview of applications of minimal uscos/cuscos: 1. 2. 3. 4.

the study of differentiability of Lipschitz functions, [4, 36, 44]; selection theorems, [17, 20, 38]; the study of weak Asplund spaces, [12, 29, 39, 41, 43]; optimization, [9, 33].

In our work, we continue in the study of the topology of uniform convergence on compacta on the space of minimal usco and minimal cusco maps. In our previous papers, we proved the Arzelà-Ascoli type theorem for these spaces [22] and we also found conditions under which they are locally convex topological vector spaces [28]. We will prove that spaces of minimal usco and minimal cusco maps from a locally compact space to a Fréchet space, equipped with the topology of uniform convergence on compacta are isomorphic as topological vector spaces. Moreover, if the domain space is also hemicompact, both spaces are Fréchet. In this chapter, we will often use characterizations of minimal usco and minimal cusco maps via their selections [22, 28].

1.2 Minimal Usco and Minimal Cusco Maps Throughout the chapter, let X, Y be Hausdorff topological spaces; let Z+ be the set of positive integers, R the space of real numbers, and C the space of complex numbers, with their usual metrics. By a topological vector space, we mean a topological vector space over the field K, which is either R or C. The symbols A and Int(A) will stand for the closure and the interior of the set A in a topological space. If Y is a topological vector space and A ⊆ Y , denote by coA the convex hull of A and its closure as co A, i.e., co A = coA. A multifunction, or a set-valued map, from X to Y is a function that assigns to each element of X a subset of Y . Following [11], the term map is reserved for a set-valued map. Denote by S(F) the set of points of X , where a map F is singlevalued; i.e., |F(x)| = 1. If F is a map from X to Y , then its graph is the set {(x, y) ∈ X × Y : y ∈ F(x)}. Throughout the chapter, we will identify maps with their graphs. If f : X → Y is a single-valued function, we will use the symbol f also for the graph of f . Notice that we can consider it to be a special case of a set-valued map that fulfills S( f ) = X . The symbol f denotes the closure of the graph of f in X × Y . In general, it is a set-valued map. Let Y be a topological vector space and F be a map from X to Y . Denote by coF a map (from X to Y ) defined as (coF)(x) = co[F(x)]. A map F : X → Y is upper semicontinuous at a point x ∈ X if for every open set V containing F(x), there exists an open set U such that x ∈ U and F(U ) =

 {F(u) : u ∈ U } ⊆ V.

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F is upper semicontinuous if it is upper semicontinuous at each point of X . We say that a map F is usco [8] if it is upper semicontinuous and has non-empty compact values. A map F from a topological space X to a topological vector space Y is cusco if it is usco and F(x) is convex for every x ∈ X . Important classes of usco maps are minimal usco and minimal cusco maps. A map F from a topological space X to a topological (topological vector) space Y is said to be minimal usco (minimal cusco) if it is a minimal element in the family of all usco (cusco) maps from X to Y ; that is, if it is usco (cusco) and does not contain properly any other usco (cusco) map from X into Y . It is known that every usco (cusco) map from X to Y contains a minimal usco (cusco) map from X to Y (see [6, 7, 11]). Denote by MU (X, Y ) and MC(X, Y ) the space of all minimal usco and minimal cusco maps from X to Y , respectively. In [32, 35], another approach to minimality can be found. The papers [22, 28] contain interesting characterizations of minimal usco maps using quasicontinuous and subcontinuous selections. A function f : X → Y is quasicontinuous at x ∈ X [40] if for every neighborhood V of f (x) and every neighborhood U of x there is a non-empty open set G ⊆ U such that f (G) ⊆ V . If f is quasicontinuous at every point of X , we say that f is quasicontinuous. The notion of quasicontinuity was perhaps the first used by Baire in [2] to study points of separately continuous functions. There is a rich literature concerning the study of quasicontinuity; see [2, 18, 31, 32, 40]. A function f : X → Y is subcontinuous at x ∈ X [14], if for every net (xi ) convergent to x, there is a convergent subnet of ( f (xi )). If f is subcontinuous at every x ∈ X , we say that f is subcontinuous. The notion of subcontinuity can be extended for so-called densely defined functions. Let A be a dense subset of a topological space X , and Y be a topological space. Let f : A → Y be a function. We say that f is densely defined. Further we say that f : A → Y is subcontinuous at x ∈ X [34], if for every net (xi ) ⊆ A convergent to x ∈ X , ( f (xi )) has a convergent subnet. A function f : A → Y is subcontinuous on X if it is subcontinuous at every x ∈ X . If A is a dense subset of X , a continuous function f : A → Y is both subcontinuous and quasicontinuous on A. However, it may fail to be subcontinuous on X . A function f : A → Y is a dense selection of a set-valued map F from X to Y , iff A is dense in X and for every x ∈ A it holds f (x) ∈ F(x). The following theorems collect several characterizations of minimal usco maps from [17, 20, 22, 28]. Theorem 1 Let X, Y be topological spaces, Y be regular, and let F be a map from X to Y . Then the following are equivalent: 1. F ∈ MU (X, Y ); i.e. it is minimal usco, 2. F has a selection f : X → Y that is both subcontinuous and quasicontinuous on X , and f = F, 3. F has a dense selection f : D → Y that is subcontinuous on X , quasicontinuous on D, and f = F, 4. every selection f : X → Y of F is both subcontinuous and quasicontinuous on X , and f = F,

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5. every dense selection f : D → Y of F is subcontinuous on X , quasicontinuous on D, and f = F, 6. every dense selection f : D → Y of F is subcontinuous on X and f = F. Proof The equivalence of (1)–(4) follows from [22, Theorem 2.2]. The equivalence of (4) and (5) is trivial as well as the implication (5) ⇒ (6). To prove (6) ⇒ (1) take a selection h : X → Y of F. By the assumption, h is subcontinuous and h = F. By [27, Corollary 3.12], F is usco. To prove that F is minimal usco, take an arbitrary usco G ⊆ F and g a dense selection of G, then g ⊆ G. From the assumption, we have that F = g ⊆ G; thus, F has to be minimal.  Corollary 1 Let X, Y be topological spaces, Y be regular, and let F, G ∈ MU (X, Y ). If the set {x ∈ X : F(x) = G(x)} is dense in X , then F = G. Proof Choose a selection f of F ∩ G. Then f is a dense selection of both F and G and by Theorem 1 we have F = f = G.  The following theorem characterizes minimal cusco maps analogical to the Theorem 1. One can easily verify (see Example 1) that not every selection of a minimal cusco map needs to be quasicontinuous; thus, we will need a somewhat weaker notion. The following definitions and facts are from [37]. Let F be a map from a topological space X to a topological space Y . F is called minimal, if for every open set W ⊆ Y and an open set U ⊆ X with F(U ) ∩ W = ∅ there is a non-empty open subset V ⊆ U such that F(V ) ⊆ W . Note that a function from X to Y is minimal iff it is quasicontinuous on X . An usco map is minimal usco, iff it is minimal. Let F be a map from a topological space X to a topological vector space Y . F is hyperplane minimal, if for every open half-space W ⊆ Y and an open set U ⊆ X with F(U ) ∩ W = ∅ there is a non-empty open subset V ⊆ U such that F(V ) ⊆ W . A cusco map is minimal cusco iff it is hyperplane minimal. Note that every quasicontinuous function is hyperplane minimal, but not every hyperplane minimal function is quasicontinuous, see Example 1. Lemma 1 ([19]) Let X be a topological space and Y be a locally convex topological vector space. Suppose a map F from X to Y is usco and coF has compact values, then coF is cusco. Theorem 2 Let X be a topological space, Y be a locally convex topological vector space, and let F be a map from X to Y with compact values. Then the following are equivalent: 1. F ∈ MC(X, Y ); i.e. it is minimal cusco, 2. F has a selection f : X → Y that is both subcontinuous and quasicontinuous on X , and F = co f , 3. F has a dense selection f : D → Y that is subcontinuous on X , quasicontinuous on D, and F = co f ,

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4. every selection f : X → Y of F is both subcontinuous and hyperplane minimal on X and co f = F, 5. every dense selection f : D → Y of F is subcontinuous on X , hyperplane minimal on D, and co f = F, 6. every dense selection f : D → Y of F is subcontinuous on X and F = co f . Proof The equivalence of (1)–(3) follows from [22, Theorem 2.3]. The equivalence of (1) and (4) is from [21, Theorem 5.4] and (4) ⇔ (5) is trivial as well as the implication (5) ⇒ (6). To prove (6) ⇒ (1) take a selection h : X → Y of F. Then h is subcontinuous, and h is usco, by the argument similar to the proof of Theorem 1. By Lemma 1 coh = F is cusco. To prove that F is minimal cusco, take an arbitrary cusco G ⊆ F and g a dense selection of G. Then co g ⊆ G. From the assumption, we have that  F = co g ⊆ G; thus, F has to be minimal. Note that Theorems 1 and 2 are not entirely symmetric. Since coK , for a compact set K does not need to be compact, we have to assume compact values of F in Theorem 2. In spaces where the closed convex hull of a compact set is compact, we may omit this assumption. There are three important cases when the closed convex hull of a compact set is compact. The first is when the compact set is a finite union of compact convex sets. The second is when the space is completely metrizable and locally convex. This includes the case of all Banach spaces with their norm topologies. The third case is a compact set in the weak topology on a Banach space; see [1]. Corollary 2 Let X be a topological space, Y be a locally convex topological vector space, and let F, G ∈ MC(X, Y ). If the set {x ∈ X : F(x) = G(x)} is dense in X , then F = G. Proof The proof is analogical to the one of Corollary 1.



Continuous functions, when viewed as set-valued maps, are both minimal usco and minimal cusco. In fact, both spaces MU (X, Y ) and MC(X, Y ) are, in the sense of the above theorems, generalizations of the space of continuous functions. We will present some illustrative examples. Example 1 Let f, h : R\{0} → R be given by f (x) = sgn(x) and h(x) = sin x1 , let g : R → R be given by g(x) = sgn(x) and let F, G, H ∈ M(R, R) be defined as follows: ⎧ ⎧  ⎪ ⎪ x < 0, x < 0, ⎨{−1}, ⎨{−1}, {sin x1 }, x = 0, F(x) = {−1, 1}, x = 0, G(x) = [−1, 1], x = 0, H (x) = ⎪ ⎪ [−1, 1], x = 0. ⎩ ⎩ {1}, x > 0. {1}, x > 0.

Then f, h are continuous on R\{0} and subcontinuous on R. Note that g is subcontinuous and hyperplane minimal on R but not quasicontinuous on R. Also F = f ∈ MU (R, R), G = co f ∈ MC(R, R) and H = h = coh ∈ MU (R, R) ∩ MC(R, R). Their graphs are in Fig. 1.1.

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(c) Map H, that is both minimal usco and minimal cusco.

Fig. 1.1 Maps from the Example 1

1.3 When Is the Space of Minimal USCO/CUSCO Maps a Completely Metrizable Space? Let X be a topological space and (Y, d) be a metric space. The open d-ball with center z 0 ∈ Y and radius ε > 0 will be denoted by Sε (z 0 ) and the ε-parallel body  a∈A Sε (a) for a subset A of Y will be denoted by Sε (A). By Bε (A) denote the set {y ∈ Y : d(y, A) ≤ ε}. Denote by C L(Y ) the space of all non-empty closed subsets of Y and by K (Y ) the space of all non-empty compact subsets of Y . If A ∈ C L(Y ), the distance functional d(., A) : Y → [0, ∞) is described by the familiar formula d(z, A) = inf{d(z, a) : a ∈ A}. Let A and B be non-empty subsets of (Y, d). The excess of A over B with respect to d is defined by the formula ed (A, B) = sup{d(a, B) : a ∈ A}. Following [3], the Hausdorff (extended-valued) metric Hd on C L(Y ) is defined by Hd (A, B) = max{ed (A, B), ed (B, A)}.

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We will often use the following equality on C L(Y ): Hd (A, B) = inf{ε > 0 : A ⊆ Sε (B) and B ⊆ Sε (A)}. The topology generated by Hd is called the Hausdorff metric topology. Following [15] we will define several topologies on Z := C L(Y ) X , the space of maps from X to Y with non-empty closed values. For a set K ⊆ X and ε > 0 let us define W (K , ε) = {(F, G) ∈ Z 2 : Hd (F(x), G(x)) < ε, for every x ∈ K }. The topology of pointwise convergence, τ p , is induced by the uniformity U p , which has the following base {W (A, ε) : A is finite and ε > 0}. The topology of uniform convergence, τU , is induced by the uniformity UU which has the following base {W (X, ε) : ε > 0}. Note that the uniformity UU has a compatible (possibly infinite-valued) metric σ defined by σ (F, G) = sup{Hd (F(x), G(x)) : x ∈ X )}. The topology of uniform convergence on compact sets, τU C , is induced by the uniformity UU C , which has the following base {W (K , ε) : K is compact and ε > 0}. The general τU C -basic neighborhood of F ∈ Z will be denoted by W (F, K , ε), i.e. W (F, K , ε) = W (K , ε)[F]. We will be interested mainly in the topology τU C on the space MU (X, Y ) of minimal usco maps from a topological space X to a metric space (Y, d) and on the space MC(X, Y ) of minimal cusco maps from a topological space X to a metric vector space (Y, d). Some topological properties of the space (MU (X, Y ), τU C ) can be found in [16, 26]. Cardinal invariants of the space (MU (X, R), τU C ) are studied in [26]. We are now going to investigate when are the spaces (MU (X, Y ), τU C ) and (MC(X, Y ), τU C ) completely metrizable. Recall that a topological space X is hemicompact if there is a countable cofinal subfamily in K (X ) with respect to the inclusion. Let X be a hemicompact space and (Y, d) be a metric space. Let {K n : n ∈ Z+ } be a countable cofinal subfamily in K (X ) with respect to the inclusion. It is easy to verify that the countable family {W (K m , 1/n) : m, n ∈ Z+ } is a base of the uniformity UU C on C L(Y ) X . Thus the uniformity UU C is metrizable, see [30, Theorem 6.13]. We will define a compatible metric ρ on C L(Y ) X .

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For every K ∈ K (X ) let p K be the (extended-valued) pseudometric on C L(Y ) X defined by p K (F, G) = sup{Hd (F(x), G(x)) : x ∈ K }. Then for every K ∈ K (X ) we have the pseudometric h K defined as h K (F, G) = min{1, pk (F, G)}. We define a function ρ : C L(Y ) X × C L(Y ) X → R as follows ρ(F, G) =

∞ 1 h K (F, G). 2n n n=1

It is easy to see that ρ is a metric on C L(Y ) X that is compatible with the uniformity UU C . Notice that two different countable cofinal families in K (X ) give uniformly equivalent metrics. We have this result. Theorem 3 ([16]) Let X be a locally compact hemicompact topological space and (Y, d) be a complete metric space. The space (MU (X, Y ), τU C ) is completely metrizable by the metric ρ (defined above) restricted to MU (X, Y ). Proof If (Y, d) is a complete metric space, then (K (Y ), Hd ) is also complete, see [3]. Thus by [30, Theorem 7.10], the uniform space (K (Y ) X , UU C ) is complete too. If X is a locally compact space and (Y, d) is a metric space, then by [22], MU (X, Y ) is a closed subspace of (K (Y ) X , UU C ). Thus if X is a locally compact hemicompact space and (Y, d) is a complete metric space, then (MU (X, Y ), ρ) is a complete metric space.  We will now study the complete metrizability of minimal cusco maps. Let (Y, d) be a metric vector space. Denote by C K (Y ) the space of all compact convex sets in (Y, d). From [13, Theorem 7.4], we know that in the Hausdorff metric, the set of compact convex subsets of a Banach space Y is a complete metric space. The following result is a generalization of this. Lemma 2 Let (Y, d) be a complete metric vector space. Then (C K (Y ), Hd ) is a complete metric space. Proof Let {K n : n ∈ Z+ } be a Cauchy sequence in (C K (Y ), Hd ). Let K ∈ K (Y ) be such that {K n : n ∈ Z+ } converges to K in (K (Y ), Hd ). We prove that K is convex. Suppose there are x, y ∈ K and λ ∈ (0, 1) such that z = λx + (1 − λ)y ∈ / K . Let  > 0 be such that S [z] ∩ S [K ] = ∅. There is n 0 ∈ Z+ such that K n ⊆ S [K ] for every n ≥ n 0 . There are sequences {xn : n ∈ Z+ } and {yn : n ∈ Z+ } such that {xn , yn } ⊆ K n , {xn : n ∈ Z+ } converges to x

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and {yn : n ∈ Z+ } converges to y. Then {λxn + (1 − λ)yn : n ∈ Z+ } converges to z,  a contradiction, since λxn + (1 − λ)yn ∈ K n for every n ∈ Z+ . Using a result from [30, Theorem 7.10] we have the following fact. Proposition 1 Let X be a topological space and (Y, d) be a complete metric vector space. Then (C K (Y ) X , UU C ) is complete. Lemma 3 Let X be a locally compact topological space and (Y, d) be a locally convex complete metric vector space. Then MC(X, Y ) is a closed subspace of (C K (Y ) X , UU C ). Proof Let F ∈ MC(X, Y ) in (C K (Y ) X , UU C ). First we prove that F is upper semicontinuous. Suppose that F is not upper semicontinuous at x ∈ X . Let ε > 0 be such that for every open neighbourhood O of x there is y O ∈ F(O)\S2ε (F(x)) and B be a compact neighbourhood of x. Let G ∈ W (F, B, ε) ∩ MC(X, Y ). The upper semicontinuity of G at x implies that there is an open neighborhood U of x such that U ⊆ Int(B) and G(z) ⊆ Sε (F(x)) for every z ∈ U . Let s ∈ U be such that yU ∈ F(s)\S2ε (F(x)), a contradiction, since there must exist z ∈ G(s) such that d(yU , z) < ε. Now we prove that F is minimal cusco. Suppose that F is not minimal cusco. Since F is cusco there must exist a minimal cusco L contained in F. Let x ∈ X be such that there is y ∈ F(x)\L(x). By separating theorem [1, Corollary 5.68] there is nonzero continuous linear functional h and λ = 0 such that h(y) > λ and L(x) ⊆ {z ∈ Y : h(z) < λ}. Let ε > 0 be such that S2ε (y) ⊆ {z ∈ Y : h(z) > λ} and S2ε (L(x)) ⊆ {z ∈ Y : h(z) < λ}. Let B be a compact neighbourhood of x. Let G ∈ W (F, B, ε) ∩ MC(X, Y ). Let z ∈ G(x) ∩ Sε (y). There is a quasicontinuous selection g of G such that G = cog. There is v ∈ g(x) such that h(v) > λ (otherwise for every t ∈ g(x) we have h(t) ≤ λ and then G(x) = cog(x) ⊆ {z ∈ Y : h(z) ≤ λ}, a contradiction). Let O be an open set such that x ∈ O ⊆ Int(B) and L(s) ⊆ Sε (L(x)) for every s ∈ O. There is t ∈ O such that h(g(t)) > λ. The quasicontinuity of g at t implies that there is a non-empty open set U ⊆ O such that h(g(s)) > λ for every s ∈ U . Thus h( p) ≥ λ for every p ∈ g(s). Then we have G(s) = cog(s) ⊆ {z ∈ Y : h(z) ≥ λ} for every s ∈ U, a contradiction. Note that the above lemma was proved in [24] for Y = R.



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Corollary 3 Let X be a locally compact topological space and (Y, d) be a locally convex complete metric vector space. Then (MC(X, Y ), UU C ) is complete. Theorem 4 Let X be a locally compact hemicompact topological space and (Y, d) be a locally convex complete metric vector space. Then (MC(X, Y ), τU C ) is completely metrizable by the metric ρ (defined above) restricted to MC(X, Y ). Proof Since X is hemicompact there is a countable cofinal family {K n : n ∈ Z+ } in K (X ) with respect to the inclusion. It is easy to verify that the countable family {W (K m , 1/n) : m, n ∈ Z+ } is a base of the uniformity UU C on MC(X, Y ). Thus the uniformity UU C on MC(X, Y ) is metrizable, see [30, Theorem 6.13]. By Corollary 3, the space (MC(X, Y ), τU C ) is completely metrizable. It was mentioned above that  ρ is compatible with UU C .

1.4 Minimal USCO/CUSCO Maps with a Structure of a Vector Space Suppose that Y is a commutative topological group with group operation + and zero denoted as 0. Functions with values in Y naturally form a group too. We want to define a similar structure for set-valued maps from MU (X, Y ) or MC(X, Y ). Let F, G be arbitrary maps from X to Y . We can define F ⊕ G by (F ⊕ G)(x) = F(x) + G(x), where on the right-hand side we have standard addition of sets in a group; and also −F by (−F)(x) = −F(x). Let Θ denote the zero map from X to Y ; i.e. Θ(x) = {0}. We have the following: Proposition 2 ([28]) Let X be a topological space, Y be a commutative topological group and F, G be usco maps from X to Y , then: 1. maps −F and F ⊕ G are usco; 2. Θ ∈ MU (X, Y ); 3. if F ∈ MU (X, Y ) then −F ∈ MU (X, Y ). Moreover if Y is a topological vector space and F, G are also cusco, then: 1. maps −F and F ⊕ G are cusco; 2. Θ ∈ MC(X, Y ); 3. if F ∈ MC(X, Y ) then −F ∈ MC(X, Y ). The proof is an easy exercise using the definitions. If F, G are minimal USCO/ CUSCO, then F ⊕ G does not need to be minimal USCO/CUSCO. Take a map H from Example 1, then H, −H ∈ MU (R, R) ∩ MC(R, R), but one can easily check that  {0}, x = 0, H ⊕ (−H ) = [−2, 2], x = 0,

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is neither minimal usco nor minimal cusco. We need an addition without this issue. In [28], we have shown a meaningful definition of additions of maps from MU (X, Y ) or MC(X, Y ). We will repeat the main idea of that construction here. Recall that a topological space Y is Stegall [12] if for every Baire space X and every F ∈ MU (X, Y ), the set S(F) is residual. Any fragmentable space is a Stegall space, particularly any metric space, see [42]. Suppose that X is a Baire space and Y is a Stegall space (as well as a commutative topological group). Let F, G ∈ MU (X, Y ). Since the set S(F) ∩ S(G) is residual in X , we can choose a dense subset D in X , such that D ⊆ S(F) ∩ S(G). Define functions f, g : D → Y by f = FD , g = G D . Functions f and g are subcontinuous on X , continuous on D, and dense selections of F and G, respectively. Define a map F +u G ⊆ X × Y by F +u G = f + g. By Theorem 1 the map F +u G ∈ MU (X, Y ). Using Corollary 1, we can verify that F +u G does not depend on the particular choice of D, as long as D is dense in X . Instead of F +u (−G) we will write F −u G. Since f + g is continuous on D then S(F +u G) ⊇ D. Thus for every x ∈ D we have (F +u G)(x) = F(x) + G(x), (F −u G)(x) = F(x) − G(x), where all the above sets are singletons. Theorem 5 ([28]) Let X be a Baire space and Y be a Stegall commutative topological group. The space (MU (X, Y ), +u ) is a commutative group. Proof Commutativity of +u : Choose arbitrary F, G ∈ MU (X, Y ) and put D = S(F) ∩ S(G). The set D is dense in X and for every x ∈ D we have (F +u G)(x) = F(x) + G(x) = G(x) + F(x) = (G +u F)(x), since the equations concern only singletons. From Corollary 1 we get F +u G = G +u F. It remains to prove that for every F ∈ MU (X, Y ) we have F +u Θ = F, F −u F = Θ; and that +u is associative. This is, however, analogical to the above argument since all of these properties hold for singletons, i.e., on a dense set, and can be extended using Corollary 1.  Suppose now that X is a Baire space and Y is a Stegall locally convex topological vector space. We will show that for every F ∈ MC(X, Y ), the set S(F) is residual.

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Since F ∈ MC(X, Y ) is usco, there is a minimal usco map G such that G ⊆ F. Since F is also cusco, then coG ⊆ F. By Lemma 1 the map coG is cusco and since F is minimal cusco, F = coG. Thus S(F) = S(G), i.e. S(F) is residual; see also [25]. Suppose that F, G ∈ MC(X, Y ) and let D, f and g be defined as above. Define now a map F +c G ⊆ X × Y by F +c G = co( f + g). Observe that F +c G ⊆ F ⊕ G, thus the compactness of the set (F ⊕ G)(x) implies the compactness of the set (F +c G)(x). Using Theorem 2 we have that F +c G ∈ MC(X, Y ). Similarly to the previous case, using Corollary 2, F +c G does not depend on the particular choice of D, as long as D is dense in X . Again we will define F −c G = F +c (−G). Theorem 6 ([28]) Let X be a Baire space and Y be a Stegall locally convex topological vector space. The space (MC(X, Y ), +c ) is a commutative group. Proof The proof is similar to the proof of Theorem 5. It uses Corollary 2.



Now suppose that Y is a topological vector space over K, which is either R or C. The definition λF, the product of a set-valued map F from X to Y with a scalar λ ∈ K, is as follows: (λF)(x) = λ[F(x)], where on the right-hand side, we have a set multiplied by a scalar. The following propositions are straightforward. Proposition 3 ([28]) Let X be a topological space and Y be a topological vector space. For every λ, μ ∈ K and every map F from X to Y it holds: (λμ)F = λ(μF), 1F = F and −1F = −F. Proposition 4 ([28]) Let X be a topological space, Y be a topological vector space, F be a map from X to Y and λ ∈ K. If F is usco (resp. minimal usco, cusco, minimal cusco) then λF is usco (resp. minimal usco, cusco, minimal cusco). Theorem 7 ([28]) Let X be a Baire space and Y be a Stegall topological vector space (over K), then (MU (X, Y ), +u , ·) is a vector space (over K). Moreover if Y is locally convex, then (MC(X, Y ), +c , ·) is a vector space (over K). Proof Using the above propositions, it remains to prove the distributive laws. This is similar to the arguments in the proofs of Theorems 5 and 6.  We have found sufficient conditions under which MU (X, Y ) and MC(X, Y ) are vector spaces, so the natural question is whether they can be topological vector spaces. We can use the space C(X ) of continuous real-valued functions as an analogy. If it is equipped with τU C , it is a topological vector space. However, if it is equipped with

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the topology τU , the multiplication with the scalar can fail to be continuous. But it works for bounded continuous functions. Let F be a map from a topological space X to a topological vector space Y . We will say that F is bounded, iff F(X ) is bounded; i.e. if for every open neighborhood O of 0, there is λ ∈ R such that F(X ) ⊆ λO. Denote by B MU (X, Y ) (resp. B MC(X, Y )) the space of all bounded minimal usco (resp. cusco) maps from X to Y . The spaces B MU (X, Y ) and B MC(X, Y ) are subspaces of MU (X, Y ) and MC(X, Y ), respectively. The following theorem follows from [28, Main Theorem 7.6]. Theorem 8 Let X be a Baire space and (Y, d) be a locally convex metric vector space (over K), then 1. (MU (X, Y ), +u , τU ) and (MC(X, Y ), +c , τU ) are commutative topological groups. 2. (B MU (X, Y ), +u , ·, τU ) and (B MC(X, Y ), +c , ·, τU ) are locally convex topological vector spaces (over K). Moreover, if X is locally compact, then 3. (MU (X, Y ), +u , ·, τU C ) and (MC(X, Y ), +c , ·, τU C ) are locally convex topological vector spaces (over K); B MU (X, Y ) and B MC(X, Y ) are their respective subspaces. We get the following statement by combining the last part of the above theorem with Theorems 3 and 4. Theorem 9 Let X be a locally compact hemicompact space and (Y, d) be a locally convex complete metric vector space (over K), then (MU (X, Y ), +u , ·, τU C ) and (MC(X, Y ), +c , ·, τU C ) are completely metrizable locally convex topological vector spaces (over K).

1.5 Isomorphism of the Spaces of Minimal Usco and Minimal Cusco Maps In this part, we will study the isomorphism of MU (X, Y ) and MC(X, Y ) as topological vector spaces. Define the function ϕ : MU (X, Y ) → C L(Y ) X as follows: ϕ(F) = coF. Note that by [6, Proposition 2.7] for every F ∈ MU (X, Y ) the map ϕ(F) is minimal cusco, provided it has compact values. Theorem 10 ([19]) Let X be a topological space, Y be a locally convex topological vector space, and F ∈ MC(X, Y ). There is a unique minimal usco map contained in F. The above theorem shows that ϕ is at least partially invertible. For F ∈ MC(X, Y ) there is unique G ∈ MU (X, Y ) such that G ⊆ F. The function

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ϕ −1 : MC(X, Y ) → MU (X, Y ) is then defined as follows: ϕ −1 (F) = G. Using this, we obtain the following. Theorem 11 ([19]) Let X be a topological space and Y be a locally convex topological vector space in which the closed convex hull of a compact set is compact. The map ϕ is bijection between MU (X, Y ) and MC(X, Y ). We will need the following lemmas. Lemma 4 ([22]) Let X be a topological space and (Y, d) be a metric space. Let F, G ∈ MU (X, Y ). If ε > 0 and Hd (F(x), G(x)) ≤ ε for all x in a dense subset of X , then Hd (F(x), G(x)) ≤ ε for all x ∈ X . Lemma 5 Let (Y, d) be a locally convex metric vector space with an invariant metric d. For every ε > 0 there is δ ∈ (0, ε), such that if compact sets K , T fulfill K ⊆ Bδ (T ) and coT is compact, then coK ⊆ Bε (coT ). Proof Since Y is locally convex, then for every ε > 0 there is δ > 0 such that U := coBδ (0) ⊆ Bε (0). Suppose K , T fulfill assumptions with δ, then K ⊆ Bδ (T ) = Bδ (0) + T ⊆ U + coT. Since coT is compact convex set and U is closed convex set, then U + coT is closed convex set and coK ⊆ U + coT ⊆ Bε (0) + coT = Bε (coT ).  A locally convex topological vector space Y is called a Fréchet space [12], if its topology is induced by a complete and (translation) invariant metric. Notice that any completely metrizable locally convex topological vector space has a compatible metric d, that is complete and invariant. Thus whenever we say that (Y, d) is a Fréchet space, we assume that d is a fixed metric, with these properties. The following theorem was proved for Banach spaces Y in [19]. Theorem 12 Let X be a locally compact space and (Y, d) be a Fréchet space. Then ϕ : (MU (X, Y ), UU C ) → (MC(X, Y ), UU C ) is uniform homeomorphism. Proof Since d is complete, then for every compact C ⊆ Y , the set coC is compact. Take an arbitrary compact K ⊆ X and ε > 0. For ε/2 choose δ ∈ (0, ε/2) according to Lemma 5 and F, G ∈ MU (X, Y ) such that (F, G) ∈ W (K , δ). Then F(x) ⊆ Sδ (G(x)) ⊆ Bδ (G(x)), for every x ∈ K . From Lemma 5 we have that

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coF(x) ⊆ Bε/2 (coG(x)) ⊆ Sε (coG(x)), for every x ∈ K and the same holds for F and G swapped. Thus (ϕ(F), ϕ(G)) = (coF, coG) ∈ W (K , ε). Take an arbitrary compact K ⊆ X and ε > 0. Since X is locally compact, then there is a compact K 0 such that K ⊆ Int(K 0 ). Let F, G ∈ MC(X, Y ) be such that (F, G) ∈ W (K 0 , ε/2). Put D = S(F) ∩ S(G) ∩ Int(K 0 ) and since Int(K 0 ) is Baire and Y is Stegall, then D is dense in Int(K 0 ). For every x ∈ D we have that F(x) is singleton and since ϕ −1 (F) ⊆ F, then F(x) = ϕ −1 (F)(x). Similarly for G. Thus for every x ∈ D we have

 ε Hd ϕ −1 (F)(x), ϕ −1 (G)(x) < . 2 Since ϕ −1 (F), ϕ −1 (G) restricted to Int(K 0 ) are minimal usco, we can use Lemma 4 to obtain

 ε Hd ϕ −1 (F)(x), ϕ −1 (G)(x) ≤ < ε, 2 for every x ∈ Int(K 0 ). Finally (ϕ −1 (F), ϕ −1 (G)) ∈ W (K , ε).  Theorem 13 Let X be a Baire space and Y be a Stegall locally convex topological vector space, in which the closed convex hull of a compact set is compact. The map ϕ : MU (X, Y ) → MC(X, Y ) is linear. Proof Let F, G ∈ MU (X, Y ), λ ∈ K and D be a dense subset of S(F) ∩ S(G). Let f, g be defined as f = FD , g = G D . Notice that f, g are also dense selections of coF, coG ∈ MC(X, Y ) and thus ϕ(F +u G) = co(F +u G) = co( f + g) = (coF) +c (coG) = ϕ(F) +c ϕ(G), and ϕ(λF) = co(λF) = λcoF = λϕ(F).  Recall that a mapping between two topological vector spaces is called an isomorphism if it is a bicontinuous linear bijection. Combining Theorems 8, 9, 11, 12 and 13 we obtain the following.

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Theorem 14 Let X be a locally compact space and (Y, d) be a Fréchet space. Then (MU (X, Y ), +u , ·, τU C ) and (MC(X, Y ), +c , ·, τU C ) are isomorphic locally convex topological vector spaces, with isomorphism ϕ. Moreover if X is also hemicompact, then both MU (X, Y ) and MC(X, Y ) are Fréchet.

1.6 Conclusions The main objects considered in our chapter are the spaces of minimal usco maps (denoted by MU (X, Y )) and minimal cusco maps (denoted by MC(X, Y )) from a topological space X into the space of all nonempty compact subsets of a locally convex topological vector space. In Sect. 1.2 of the Chapter, we have collected several interesting characterizations on minimal usco and minimal cusco maps via their selections from our previous papers. In Sect. 1.3, we have studied conditions under which spaces of minimal usco and minimal cusco maps equipped with the topology τU C of uniform convergence on compacta are completely metrizable and we also defined a compatible complete metric on these spaces. In Sect. 1.4, we have defined the natural vector space structures on MU (X, Y ) and MC(X, Y ). In Sect. 1.5, we have proved that when X is a locally compact space and (Y, d) is Fréchet, then (MU (X, Y ), τU C ) and (MC(X, Y ), τU C ) with the vector space structures are isomorphic locally convex topological vector spaces. Moreover if X is also hemicompact, then both (MU (X, Y ), τU C ) and (MC(X, Y ), τU C ) are Fréchet. Concerning a future investigation of the spaces of minimal usco and minimal cusco maps, one can study topologies τB of uniform convergence on elements of B, where B is a bornology. In [28] we have studied conditions under which these spaces are topological groups and topological vector spaces. Of particular interest are metrizability, complete metrizability, and cardinal invariants of topologies τB on spaces of minimal usco and minimal cusco maps. Cardinal invariants of the topology of uniform convergence on compacta on the space of minimal usco maps were studied in [23, 26]. Acknowledgements All authors were supported by APVV-20-0045. L’. Holá and B. Novotný were supported also by VEGA 2/0048/21.

References 1. Aliprantis, C., Border, K.: Infinite Dimensional Analysis. Springer, GmbH (2006) 2. Baire, R.: Sur les fonctions de variables réelles. Ann. Mat. Pura Appl. (4), 3(1), 1–123 (1899) 3. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dodrecht (1993)

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4. Borwein, J.M.: Fixed Point Theory and Applications, volume 252 of Pitman Research Notes in Mathematics Series, Chapter Minimal Cuscos and Subgradients of Lipschitz Functions, pp. 57–81. Longman, Harlow (1991) 5. Borwein, J.M., Moors, W.B.: Essentially smooth Lipschitz functions. J. Funct. Anal. 149, 305–351 (1997) 6. Borwein, J.M., Zhu, Q.J.: Multifunctional and functional analytic techniques in nonsmooth analysis. In: Nonlinear Analysis, Differential Equations and Control, pp. 61–157. Springer (1999) 7. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer (2004) 8. Christensen, J.P.R.: Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings. Proc. Am. Math. Soc. 86(4), 649–655 (1982) ˇ 9. Coban, M.M., Kenderov, P.S., Revalski, J.P.: Generic well-posedness of optimization problems in topological spaces. Mathematika 36, 310–324 (1989) 10. Collingwood, E.F., Lohwater, A.J.: The Theory of Cluster Sets. Cambridge University Press (2004) 11. Drewnowski, L., Labuda, I.: On minimal upper semicontinuous compact-valued maps. Rocky Mt. J. Math 20(3), 737–752 (1990) 12. Fabian, M.: Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces, vol. 18. Wiley (1997) 13. Fonf, V.P., Lindenstrauss, J., Phelps, R.R.: Infinite dimensional convexity. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2 (2003) 14. Fuller, R.V.: Relations among continuous and various noncontinuous functions. Pac. J. Math. 25(3), 495–509 (1968) 15. Hammer, S.T., McCoy, R.A.: Spaces of densely continuous forms. Set-Valued Anal. 5(3), 247–266 (1997) ˇ Spaces of densely continuous forms, USCO and minimal USCO maps. Set-Valued 16. Holá, L: Anal. 11, 133–151 (2003) ˇ Holý, D.: Minimal usco maps, densely continuous forms and upper semi-continuous 17. Holá, L, functions. Rocky Mt. J. Math 39, 545–562 (2009) ˇ Holý, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces. 18. Holá, L, Rocky Mt. J. Math 41(6), 1883–1894 (2011) ˇ Holý, D.: Relations between minimal USCO and minimal CUSCO maps. Portugal. 19. Holá, L, Math. 70(3), 211–224 (2013) ˇ Holý, D.: New characterizations of minimal CUSCO maps. Rocky Mt. J. Math 44(6), 20. Holá, L, 1851–1866 (2014) ˇ Holý, D.: Minimal USCO and minimal CUSCO maps. Khayyam J. Math. 1(2), 125– 21. Holá, L, 150 (2015) ˇ Holý, D.: Minimal USCO and minimal CUSCO maps and compactness. J. Math. 22. Holá, L, Anal. Appl. 439(2), 737–744 (2016) ˇ Holý, D.: Minimal USCO maps and cardinal invariants of the topology of uniform 23. Holá, L, convergence on compacta. RACSAM 116, 27 (2022) ˇ Holý, D.: Minimal CUSCO maps and the topology of uniform convergence on com24. Holá, L., pacta. Preprint ˇ Holý, D., Moors, W.B.: USCO and Quasicontinuous Mappings. Studies in Mathe25. Holá, L., matics, vol. 81. De Gruyter (2021) ˇ McCoy, R.A.: Cardinal invariants of the topology of uniform convergence on compact 26. Holá, L, sets on the space of minimal USCO maps. Rocky Mt. J. Math 37(1), 229–246 (2007) ˇ Novotný, B.: Subcontinuity. Math. Slovaca 62, 345–362 (2012) 27. Holá, L, ˇ Novotný, B.: When is the space of minimal USCO/CUSCO maps a topological vector 28. Holá, L, space. J. Math. Anal. Appl. 489(1), 124125 (2020) 29. Kalenda, O.: Stegall compact spaces which are not fragmentable. Topol. Appl. 96(2), 121–132 (1999) 30. Kelley, J.L.: General Topology. Springer, New York (1975) 31. Kempisty, S.: Sur les fonctions quasi-continues. Fundam. Math. 19, 184–197 (1932)

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32. Kenderov, P.S., Kortezov, I.S., Moors, W.B.: Continuity points of quasi-continuous mappings. Topol. Appl. 109(3), 321–346 (2001) 33. Kenderov, P.S., Revalski, J.P.: The Banach-Mazur game and generic existence of solutions to optimization problems. Proc. Am. Math. Soc. 118(3), 911–917 (1993) 34. Lechicki, A., Levi, S.: Extensions of semicontinuous multifunctions. Forum. Math. 2(2), 341– 360 (1990) 35. Matejdes, M.: Minimality of multifunctions. Real Anal. Exch. 32(2), 519–526 (2007) 36. Moors, W.B.: A characterization of minimal subdifferential mappings of locally Lipschitz functions. Set-Valued Anal. 3, 129–141 (1995) 37. Moors, W.B., Giles, J.R.: Generic continuity of minimal set-valued mappings. J. Aust. Math. Soc. Ser. A. Pure Math. Stat. 63(2), 238–262 (1997) 38. Moors, W.B., Somasundaram, S.: USCO selections of densely defined set-valued mappings. Bull. Aust. Math. Soc. 65(2), 307–313 (2002) 39. Moors, W.B., Somasundaram, S.: A Gâteaux differentiability space that is not weak Asplund. Proc. Am. Math. Soc. 134(9), 2745–2754 (2006) 40. Neubrunn, T.: Quasi-continuity. Real Anal. Exch. 14, 259–306 (1988) 41. Preiss, D., Phelps, R.R., Namioka, I.: Smooth Banach spaces, weak Asplund spaces and monotone or USCO mappings. Isr. J. Math. 72(3), 257–279 (1990) 42. Ribarska, N.K.: Internal characterization of fragmentable spaces. Mathematika 34(2), 243–257 (1987) 43. Stegall, C.: A class of topological spaces and differentiation of functions on Banach spaces. Vorlesungen aus dem Fachbereich Mathematik der Universität Essen 10, 63–77 (1983) 44. Zajíˇcek, L.: Generic Fréchet differentiability on Asplund spaces via AE strict differentiability on many lines. J. Convex Anal. 19(1), 23–48 (2012)

Chapter 2

Contra Continuity Properties of Relations in Relator Spaces Árpád Száz

Abstract In 1994, Julian Dontchev called a function f of one topological space X to another Y to be contra continuous if, for each open subset V of Y , the inverse image f −1 [ V ] is a closed subset of X . This seems to be a rather inconvenient continuity-like property. However, despite this, it has been intensively investigated by a surprisingly great number of prominent mathematicians. Therefore, it seems reasonable to treat this notion in relator spaces having been developed by the present author and his PhD students. Namely, they provide the most convenient framework for continuity considerations. Relator space, in a narrower sense, is an ordered pair X (R) = (X, R) consisting of a set X and a family R of relations on X . Thus, it is a common generalization of ordered sets and uniform spaces. In a relator space X (R), we define some relations IntR , ClR , intR , clR , and families τR , τ-R , TR , FR , ER , DR such that, for instance, A ∈ IntR (B) if R [ A ] ⊆ B for some R ∈ R, and A ∈ ER if intR (A) = ∅. Thus, instead of contra continuous functions, for instance, we investigate a relation F on one relator space X (R) to another Y (S ) which  reverses  proximal interior in the sense that A ∈ IntR (B) implies F [ A ] ∈ ClS F [ B ] .

2.1 Introduction In [63], Dontchev called a function f of one topological space X to another Y to be contra continuous if the relation f −1 is open set reversing in the sense that for each open subset V of Y the inverse image f −1 [ V ] = {x ∈ X : f (x) ∈ V } is a closed subset of X . This seems to be a rather inconvenient continuity-like property. Namely, even the identity function of X and the composition of two contra continuous functions need not be contra continuous. However, despite this, a surprisingly great number of authors have been investigating it. Á. Száz (B) Department of Mathematics, University of Debrecen, P.O.B. 400, Debrecen 4002, Hungary e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_2

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Therefore, it seems reasonable to generalize the notion of contra continuity to relations between relator spaces. Namely, relator spaces are common generalizations of not only ordered sets and uniform spaces, but also those of proximity, closure, topological and convergence spaces. If R is a family of relations on a set X , then the family R is called a relator on X . Moreover, the ordered pair X (R) = (X, R) is called a relator space. This term was first introduced by the present author in [183, 184] by allowing R to be only a nonvoid family of reflexive relations on X . Later, it has turned out that, for several purposes such as continuity and formal context for instance, it is better to start with a family R of relations on X to another set Y [200]. Or rather with a family U of relations on P (X ) to Y which can be briefly called super relations on X to Y [168]. However, for a preliminary illustration of our present ideas, it is enough to assume now only that F is a relation on one relator space X (R) to another Y (S ). And, for instance, to say that the  relation  F is proximal interior reversing if A ∈ IntR (B) implies F [ A ] ∈ ClS F [ B ] for all A, B ⊆ X with A = ∅. Here, because of the definitions of the induced  proximal  relations IntR and ClS , and F [ A ] ∈ Cl F [ B ] mean only that R [ A ] ⊆ B the inclusions A ∈ IntR (B) S   for some R ∈ R, and S F [ A ] ∩ F [ B ] = ∅ for all S ∈ S . Thus, for instance, we shall prove the following. Theorem 1 The following assertions are equivalent : (1) F is proximal interior reversing ; (2) for each ∅ = A ⊆ X , R ∈ R and S ∈ S we have     S F [ A ] ∩ F R [ A ] = ∅ ; (3) for each ∅ = A ⊆ X and B ⊆ Y    F [ A ] ∈ IntS B) =⇒ A ∈ ClR F −1 [ B ] . Remark 1 In assertions (2) and (3), because of the corresponding definitions, we may write {x}, with x ∈ X , instead of A. Moreover, to treat contra continuity, we also say that F is topological openness reversing if A ∈ TR implies F [ A] ∈ FS. Here, A ∈ TR and F [ A ] ∈ FS mean only that A ⊆ intR (A) and clS F [ A ] ⊆ F [ A ], respectively, where for instance intR (A) = {x ∈ X : {x} ∈ IntR (A)}. Thus, by using some basic operations on relators, we shall also prove the following. Theorem 2 The following assertions are equivalent :  # (1) F ◦ R ∧ ∞ ⊆ S ∧−1 ◦ F ; (2) F is topological openness reversing ;

2 Contra Continuity Properties of Relations in Relator Spaces

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(3) for each A ⊆ X and R ∈ R ∧ there exists S ∈ S ∧ such that     S −1 F [ A ] ⊆ F R ∞ [ A ] . Remark 2 This theorem can, in principle, be derived from Theorem 1. However, in this paper, we shall only prove it with the help of an analogous theorem for proximal openness reversing relations. It will turn out that the fatness reversing relations are more important than the openness reversing ones. Namely, they have several striking characterizations. In particular, we can note that a relator space X (R) may be naturally called hyperconnected if ER ⊆ DR . That is, the identity function of it is fatness reversing. The necessary prerequisites on relations and relators, which are certainly unfamiliar to the reader, will be briefly laid out in the next preparatory sections.

2.2 A Few Basic Definitions on Relations A subset F of a product set X ×Y is called a relation on X to Y . In particular, a relation F on X to itself is simply called a relation on X . And,  X = {(x, x) : x ∈ X } is called the identity relation of X . If F is a relation on X to Y , then by the above definitions we can also state that F is a relation on X ∪ Y . However, the latter view of the relation F would be quite unnatural for several purposes. If F is a relation on X to Y , then for  any x ∈ X and A ⊆ X the sets F (x) = {y ∈ Y : (x, y) ∈ F} and F [ A ] = a∈A F (a) are called the images or neighbourhoods of x and A under F, respectively. If (x, y) ∈ F, then instead of y ∈ F (x), we may also write x F y. However, instead of F [ A ], we cannot write F (A). Namely, it may occur that, in addition to A ⊆ X , we also have A ∈ X . The sets D F = { x ∈ X : F (x) = ∅ } and R F = F [ X ] are called the domain and range of F, respectively. If in particular D F = X , then we say that F is a relation of X to Y , or that F is a total (or non-partial) relation on X to Y . If F is a relation on X to Y and U ⊆ D F , then the relation F | U = F ∩ (U ×Y ) is called the restriction of F to U . Moreover, if F and G are relations on X to Y such that D F ⊆ DG and F = G | D F , then G is called an extension of F. In particular, a relation f on X to Y is called a function if for each x ∈ D f there exists y ∈ Y such that f (x) = {y}. In this case, by identifying singletons with their elements, we may simply write f (x) = y instead of f (x) = {y}. Moreover, a function  of X to itself is called a unary operation on X . A function ∗ of X 2 to X is called a binary operation on  X . And,  for any x, y ∈ X , we usually write x  and x ∗ y instead of (x) and ∗ (x, y) .

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If F is a relation on X to Y , then a function f of D F to Y is called a selection function of F if f (x) ∈ F (x) for all x ∈ D F . Thus, by the Axiom of Choice, we can see that every relation is the union of its selection functions.  If F is a relation on X to Y , then we can easily see that F = x∈X {x}× F(x). Therefore, the values F(x), where x ∈ X , uniquely determine F. Thus, a relation F on X to Y can also be naturally defined by specifying F(x) for all x ∈ X . For instance, the complement F c and the inverse F −1 can be defined such that c F (x) = Y \F(x) for all x ∈ X and F −1 (y) = {x ∈ X : y ∈ F(x)} for all y ∈ Y . Thus, we also have F c = X ×Y \F and F −1 = {(y, x) ∈ Y × X : (x, y) ∈ F}. Moreover, if in addition G is a relation on Y to Z , then the composition G ◦ F can be defined such that ( G ◦ F )(x) = G [ F(x) ] for all x ∈ X . Thus, we also have G ◦ F = {(x, z) ∈ X × Z : ∃ y ∈ Y : (x, y) ∈ F, (y, z) ∈ G}. While, if G is a relation on Z to W , then the box product F  G can be naturally defined such that (F  G)(x, z) = F(x) × G(z) for all x ∈ X and z ∈ Z . Thus, in particular ( X  F)(x, x) = {x} × F(x) for all x ∈ X . For any relation F on X to Y , we may naturally define two set-valued functions ϕ F of X to P(Y ) and  F of P(X ) to P(Y ) such that ϕ F (x) = F(x) for all x ∈ X and  F ( A ) = F [ A ] for all A ⊆ X . Functions of X to P(Y ) can be identified with relations on X to Y . While, functions of P(X ) to P(Y ) are more general objects than relations on X to Y . They were briefly called corelations on X to Y in [215, 219, 221]. However, if F is a relation on X to Y , G is a relation on P(X ) to Y , and H is a relation on P(X ) to P (Y ), then it is better to say that F is an ordinary relation, G is a super relation, and H is a hyper relation on X to Y [168]. Namely, thus a super relation on X to Y is an arbitrary subset of P (X )× Y . While, a corelation on X to Y is a particular subset of P(X )× P(Y ). Thus, set inclusion is a natural partial order for super relations, but not for corelations. Now, a relation R on X may be briefly defined to be reflexive if  X ⊆ R, and transitive if R ◦ R ⊆ R. Moreover, R may be briefly defined to be symmetric if R −1 ⊆ R, and antisymmetric if R ∩ R −1 ⊆  X . Thus, a reflexive and transitive (symmetric) relation may be called a preorder (tolerance) relation. And, a symmetric (antisymmetric) preorder relation may be called an equivalence (partial order) relation. For any relation R on X , we may also defineR 0 =  X , and R n = R ◦ R n−1 n if n ∈ N. Moreover, we may also define R ∞ = ∞ n=0 R . Thus, it can be shown ∞ that R is the smallest preorder relation on X containing R [91]. Now, in contrast to ( R c )c = R and ( R −1 )−1 = R, we have ( R ∞ )∞ = R ∞ . Moreover, analogously to ( R c )−1 = ( R −1 )c , we also have ( R ∞ )−1 = ( R −1 )∞ . Thus, in particular R −1 is also a preorder on X if R is a preorder on X . For A ⊆ X , the Pervin relation R A = A2 ∪ ( Ac × X ) is an important preorder on X [160]. While, for a pseudometric d on X , the Weil surrounding Br = {(x, y) ∈ X 2 : d(x, y) < r }, with r > 0, is an important tolerance on X [237].  c 2 2 It can be easily seen that S A = R A ∩ R −1 A = R A ∩ R Ac = A ∪ A ) is already an equivalence  relation on X . And, more generally if A is a cover (partition) of X , then SA = A∈A A2 is a tolerance (equivalence) relation on X .

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As an important generalization of the Pervin relation R A , for any A ⊆ X and B ⊆ Y , we may also naturally consider the Hunsaker-Lindgren relation R(A,B) = (A× B) ∪ (Ac ×Y ) [94]. Namely, thus we evidently have R A = R(A,A) . The Pervin relations R A and the Hunsaker-Lindgren relations R(A,B) were actually first used by Davis [60] and Császár [57, pp. 42 and 351] in some less explicit and convenient forms, respectively.

2.3 A Few Basic Definitions on Relators A family R of relations on one set X to another  Y is called a relator on X to Y , and the ordered pair (X, Y )(R) = (X, Y ), R is called a relator space. For the origins of this notion, see [184, 200], and the references in [184]. If in particular R is a relator on X to itself, then R is simply called a relator on X . Thus, by identifying singletons with their elements, we may naturally write X (R) instead of (X, X )(R). Namely, (X, X ) = {{X }, {X, X }} = {{X }}. Relator spaces of this simpler type are already substantial generalizations of the various ordered sets [59, 217] and uniform spaces [85, 104]. However, they are insufficient for some important purposes. (See, [86] and [200, 213, 216, 218, 225].) A relator R on X to Y , or the relator space (X, Y )(R), is called simple if R = {R} for some relation R on X to Y . Simple relator spaces (X, Y )(R) and X (R) were called formal contexts and gosets in [86] and [217], respectively. Moreover, a relator R on X , or the relator space X (R), may, for instance, be naturally called reflexive if each member of R is reflexive on X . Thus, we may also naturally speak of preorder, tolerance and equivalence relators. For instance, for a family A of subsets of X , the family RA = { R A : A ∈ A }, where R A = A2 ∪ (Ac × X ), is an important preorder relator on X . Such relators were first used by Pervin [160] and Levine [120]. While, for a family D of pseudo-metrics on X , the family RD = {Brd : r > 0, d ∈ D }, where Brd = {(x, y) : d(x, y) < r }, is an important tolerance relator on X . Such relators were first considered by Weil [237]. Moreover, if S is a family of  covers (partitions) of X , then the family RS = {SA : A ∈ S}, where SA = A∈A A2 , is an important tolerance (equivalence) relator on X . Equivalence relators were first studied by Levine [119]. If  is a unary operation for relations on X to Y , then for any relator R on X to  Y we may naturally define R  = R  : R ∈ R . However, this plausible notation may cause confusions if  is a set-theoretic operation. For instance, for any relator R on X to Y , we may naturally define the elementwise complement R c = {R c : R ∈ R}, which may easily be confused with the global complement R c = P (X ×Y )\R of R. However, for instance, the practical notations R −1 = {R −1 : R ∈ R} and ∞ R = {R ∞ : R ∈ R}, whenever R is only a relator on X , will certainly not cause confusions in the sequel.

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 In particular, for a relator R on X , we may also naturally define R ∂ = S ⊆ X 2 : S ∞ ∈ R . Namely, for any two relators R and S on X , we evidently have R ∞ ⊆ S ⇐⇒ R ⊆ S ∂ . That is, ∞ and ∂ form a Galois connection [59]. The operations ∞ and ∂ were first introduced by Mala [123, 125] and Pataki [158, 159], respectively. These two former PhD students of mine, together with János Kurdics [112, 115], made significant developments in the theory of relators. Moreover, if ∗ is a binary operation forrelations, then for any two relators R and S we may naturally define R ∗ S = R ∗ S : R ∈ R, S ∈ S . However, this notation may again cause confusions if ∗ is a set-theoretic operation. Therefore, in our former papers, we defined R ∧ S = { R ∩ S : R ∈ R, S ∈ S }. Moreover, for instance, we also defined RR −1 = R ∩ R −1 : R ∈ R . Thus, RR −1 is a symmetric relator such that RR −1 ⊆ R ∧ R −1 . A function  of the family of all relators on X to Y is called a direct (indirect) unary operation for relators if, for every relator R on X to Y , the value R  =  (R) is a relator on X to Y (on Y to X ). More generally, a function F of the family of all relators on X to Y is called a structure for relators if, for every relator R on X to Y , the value FR = F (R) is in a power set depending only on X and Y . Concerning structures and operations for relators, we may freely use some basic terminology on corelations (set-to-set functions) [221], or more generally that on functions on one poset (partially ordered set) to another [217]. For instance, the structure F is called increasing if R ⊆ S implies FR ⊆ FS for any two relators R and S on X to Y . And, F is called quasi-increasing if F R = F{R} ⊆ FR for any relator R on X to Y and R ∈ R. Moreover, the operation  is called extensive, intensive, involutive and idempotent if for any relator R on X to Y we have R ⊆ R  , R  ⊆ R, R   = R and R   = R  , respectively. In particular, an increasing involutive (idempotent) operation for relators is called an involution (projection or modification) operation. An extensive projection operation for relators is called a closure or refinement operation. Moreover, an increasing extensive operation may be called a preclosure operation. And, an extensive idempotent operation may be called a semiclosure operation. The corresponding interior operations can be defined quite similarly. For instance, c and −1 are involution operations for relators. ∞ and ∂ are projection operations for relators. Moreover, the operation  = c, ∞ or ∂ is inversion −1  −1   = R . compatible in the sense that R  While, if for instance intR (B) = {x ∈ X : ∃ R ∈ R : R (x) ⊆ B} for every relator R on X to Y and B ⊆ Y , then the function F, defined by F (R) = intR , is a union-preserving structure for relators. The first basic problem in the theory of relators is that, for an increasing structure F, we have to find an operation  for relators such that, for any two relators R and S on X to Y we could have FS ⊆ FR ⇐⇒ S ⊆ R  .

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By using Pataki connections [158, 223], several closure operations can be derived from union-preserving structures. However, more generally, one can find first the Galois adjoint G of such a structure F, and then take F = G ◦ F [204]. By finding the Galois adjoint of the structure F, the second basic problem for relators, that which structures can be derived from relators, can also be solved. However, for this, some direct methods can also be well used [191, 206]. Now, for an operation  for relators, a relator R on X to Y may be naturally called –fine if R  = R. And, for some structure F for relators, two relators R and S on X to Y may be naturally called F–equivalent if FR = FS . Moreover, for a structure F for relators, a relator R on X to Y may, for instance, be naturally called F–simple if FR = F R for some relation R on X to Y . Thus, in particular singleton relators have to be actually called properly simple. Finally, we note that a relator R on X to Y will be called non-partial if each member of R is non-partial. Moreover, the relator R, or the relator space (X, Y )(R), will be called non-degerated if both X and R are nonvoid.

2.4 A Few Basic Theorems on Relations and Relators Concerning relations and relators, we shall frequently use the subsequent simple, but important theorems whose proofs will be left to the reader. Theorem 3 For any two relations F and G on X to Y , the following assertions are equivalent : (1) F ⊆ G ; (2) F(x) ⊆ G( x) for all x ∈ X . Remark 3 Thus, if in particular F(x) = G(x) for all x ∈ X , then F = G. Theorem 4 If F is a relation on X to Y , then for any A ⊆ X and B ⊆ Y , the following assertions are equivalent : (1) F [ A ] ∩ B = ∅ ; (2) A ∩ F −1 [ B ] = ∅. Remark 4 Thus,  in particular  (1) F −1 [ B ] = x ∈ X : F (x) ∩ B = ∅ ; (3) D F −1 = F [ X ] = R F . (2) D F = F −1 [ X ] = R F −1 ; Theorem 5 If F and G are relations on X to Y , then  −1  −1 c (1) F c = F ; (2) (F\G)−1 = F −1 \G −1 . Theorem 6 If F and G are relators on X to Y , then (1)



F

−1

=

F∈F

F −1 ;

(2)



F

−1

=

 F∈F

F −1 ;

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Á. Száz



Fc

−1

c  = F −1 ;

(4) (F \G )−1 = F −1 \G −1 .

  Remark 5 Thus, both the elementwise complement F c = F c : F ∈ F and   the global complement F c = P X ×Y \F of F are inversion compatible. Theorem 7 If F is a relation on X to Y and G is a relation on Y to Z , then for any A ⊆ X we have   (G ◦ F) [ A ] = G F [ A ] . Remark 6 In particular, we have (1) F ◦ ∅ = ∅ = ∅ ◦ F ;

(2) F ◦ X 2 = X × R F ;

(3) Y 2 ◦ F = D F ×Y ;

(4) F ◦  X = F = Y ◦ F.

Theorem 8 If F is a relation on X to Y and G is a relation on Y to Z , then (G ◦ F)−1 = F −1 ◦ G −1 . Theorem 9 If in addition to the assumptions of the above theorem H is a relation on Z to W , then H ◦ (G ◦ F) = (H ◦ G) ◦ F. Remark 7 Hence, by Remark 6, we can see that P(X 2 ), with composition, is a particular monoid (semigroup with identity). Therefore, for any relation R on X , we could define R 0 =  X and ∞naturally n n−1 ∞ n if n ∈ N, and thus also R = n=0 R . R = R◦R Theorem 10 If R is a relation on X , then R ∞ is the smallest preorder relation on X such that R ⊆ R ∞ . Hint A detailed investigation of the operation ∞ can be found in the paper [91] of my former PhD student Tamás Glavosits. Remark 8 From Theorem 10, it is clear that R is a preorder on X if and only if R = R∞. Theorem 11 For any relation R on X , we have  ∞ (1) R ∞ = R ∞ ;

(2)



R∞

−1

 ∞ = R −1 .

Remark 9 Thus, in particular R −1 is a preorder on X if and only if R is a preorder on X . Most of the latter results can also be naturally extended to relators. Moreover, one can also easily check the following.

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Theorem 12 For any two relators R and S on X , we have R∞ ⊆ S

⇐⇒

R ⊆ S ∂.

Remark 10 Therefore, the operations ∞ and ∂ establish a Galois connection by [59, p. 55]. Thus, in particular we can note that ∞ ∂ is a closure operation for relators such that ∞ = ∞ ∂∞. Theorem 13 For a relation F on X to Y , the following assertions are equivalent : (1) F is total ;

(2)  X ⊆ F −1 ◦ F ;

(3) F [ A ] ⊆ B implies A ⊆ F −1 [ B ] for all A ⊆ X and B ⊆ Y . Remark 11 Thus, F is total if and only if F −1 ◦ F is reflexive on X . Theorem 14 For a relation F on X to Y , the following assertions are equivalent : (1) F is a function ;

(2) F ◦ F −1 ⊆ Y ;

(3) A ⊆ F −1 [ B ] implies F [ A ] ⊆ B for all A ⊆ X and B ⊆ Y . Remark 12 Thus, in particular F is a function if and only if F ◦ F −1 =  R F . Theorem 15 For a function f on X to Y and a function g on Y to X , the following assertions are equivalent : (1) g = f −1 ;

(2)

f ◦ g =  Dg and g ◦ f =  D f .

Theorem 16 If f is a function of X to Y , then for any A ⊆ X and B ⊆ Y we have f [ A ] ⊆ B ⇐⇒ A ⊆ f −1 [ B ]. Remark 13 Thus, the corelations  f and  f −1 establish a Galois connection between the posets P(X ) and P (Y ). While, if F is a relation on X to Y , then by Theorem 4 we can also state that  F and  F −1 also establish a Galois connection between P(X ) and P (Y ) if we write A ≤ B whenever A ∩ B = ∅. Remark 14 Galois connections occur in almost every branches of mathematics. They allow of transposing notions and statements from one world of our imagination to another one. ( For their extensive theories, see [32, 59, 61, 90].) Some examples, applications and generalizations of Galois connections can also be found in our former papers [204, 205, 208, 211, 215, 216, 220–223]. However, it is usually enough to consider such connections only for corelations.

28

Á. Száz

2.5 Some Further Theorems on Relations and Relators Theorem 17 If F is a relation on X to Y , then for any A, B ⊆ X we have (1) F [ A ]\F [ B ] ⊆ F [ A\B ] ;

(2) F [ A ]c ⊆ F [ Ac ] if Y = R F .

Theorem 18 If F is a relation on X to Y , then for any family A of subsets of X we have (1) F



A



⊆



F [ A ];

(2) F



A∈A

A



=



F [ A ].

A∈A

Remark 15 If in particular F −1 is a function ( i. e., F (x) ∩ F (z) = ∅ implies x = z for all x, z ∈ X ), then the corresponding equalities are also true in the above two theorems. Theorem 19 If F and G are relations on X to Y , then for any A ⊆ X we have (1) F [ A ]\G [ A ] ⊆ (F\G) [ A ] ;

(2) F [ A ]c ⊆ F c [ A ] if A = ∅.

Theorem 20 If F is a relator on X to Y , then for any A ⊆ X we have (1)



 F [ A ]; F [ A] ⊆

(2)



  F [ A] = F [ A ].

F∈F

F∈F

Remark 16 If in particular A is a singleton, then the corresponding equalities are also true in the above two theorems. Theorem 21 If F is a relation on X to Y , the for any A ⊆ X we have F c [ A ]c =



F(x).

x∈A

Corollary 1 If F is a relation on X to Y and G is a relation on Y to Z , then for any x ∈ X we have  G ◦ F)c (x) = G c (y). y∈F(x)

Remark 17 Thus, if in particular f is a function on X to Y and G is a relation on Y to Z , then ( G ◦ f )c = G c ◦ f. Theorem 22 If F is a relation on X to Y and G is a relation on Y to Z , then (1) (G ◦ F)c ⊆ G c ◦ F if X = D F ; (2) (G ◦ F)c ⊆ G ◦ F c if Z = RG .

2 Contra Continuity Properties of Relations in Relator Spaces

29

Theorem 23 If F and G are relations on X to Y and H is a relation on Y to Z , then       H ◦ F \ H ◦ G ⊆ H ◦ F\G . Theorem 24 If F is a relation on X to Y and G and H are relations on Y to Z , then       G ◦ F \ H ◦ F ⊆ G\H ◦ F. Theorem 25 If F is a relator on X to Y and G is a relation on Y to Z , then   G ◦ F; (2) G ◦ F = G ◦ F. (1) G ◦ F ⊆ F∈F

F∈F

Theorem 26 If F is a relation on X to Y and G is a relator on Y to Z , then      (1) G ◦F⊆ G ◦ F; (2) G ◦F= G ◦ F. G∈G

G∈G

Theorem 27 For any relations F on X to Z and G on Y to W , we have ( F  G )−1 = F −1  G −1 . Theorem 28 If F is a relation on X to Z and G is a relation on Y to W , then for any R ⊆ X ×Y we have ( F  G ) [ R ] = G ◦ R ◦ F −1 . Hint The proofs of the latter two theorems, and some applications of the box product of relations, can be found in our former paper [214, 220]. Corollary 2 For any relations F on X to Y and G on Y to Z , we have   G ◦ F = F −1  G [ Y ]. Corollary 3 If F is a relation on X to Z and G is a relation on Y to W , then for any x ∈ X and y ∈ Y , we have ( F  G )(x, y) = G ◦ {(x, y)} ◦ F −1 . Remark 18 The above two corollaries show that the box and composition products of two relations are actually equivalent tools. However, in contrast to the composition product, the box product of relations can be immediately defined for arbitrary families of relations.

30

Á. Száz

From Theorem 28, by using Theorem 27, we can also easily derive Theorem 29 If F is a relation on X to Z and G is a relation on Y to W , then for any S ⊆ Z ×W we have ( F  G )−1 [ S ] = G −1 ◦ S ◦ F. Remark 19 Note that if F is a relator on X to Z and G is a relator on Y to W , then analogously to the composition of relators we may also naturally define  F G = F G :

 F ∈ F, G ∈ G .

Therefore, Theorems 27, 28 and 29 can also be easily extended to relators.

2.6 Some Basic Structures Derived from Relators Notation 1 In this section, we shall assume that R is a relator on X to Y . Definition 1 For any A ⊆ X and B ⊆ Y , we write (1) A ∈ IntR (B) if R [ A ] ⊆ B for some R ∈ R ; (2) A ∈ ClR (B) if R [ A ] ∩ B = ∅ for all R ∈ R. The relations ClR and IntR are called the proximal closure and proximal interior generated by R, respectively. Remark 20 The origins of the relations ClR and IntR go back to Efremoviˇc’s proximity δ [66] and Smirnov’s strong inclusion  [181], respectively. The present more convenient notations were first introduced in our former paper [184], where unfortunately only nonvoid, reflexive relators were considered. The forthcoming simple, but important theorems have been mostly proved in our former papers. Therefore, their proofs will not be included here. Theorem 30 For any B ⊆ Y , we have (1) ClR (B) = P(X )\IntR ( Y \B); (2) IntR (B) = P(X )\ClR ( Y \B). Remark 21 By using appropriate complementations, assertion (1) can be written in the more concise forms c c   (2) ClR = IntR ◦ CY . (1) ClR = IntR ◦ CY ;

2 Contra Continuity Properties of Relations in Relator Spaces

31

Theorem 31 We have (1) ClR −1 = ClR−1 ;

(2) IntR −1 = CY ◦ IntR−1 ◦ C X .

Theorem 32 We have (1) ClR (∅) = ∅ and Cl−1 R (∅) = ∅ if R  = ∅ ; −1 (2) ClR (B1 ) ⊆ ClR (B2 ) if B1 ⊆ B2 ⊆ Y and Cl−1 R (A1 ) ⊆ ClR (A2 ) if A1 ⊆ A2 ⊆ X . Theorem 33 We have (1) IntR (X ) = P(X ) and Int−1 R (∅) = P(Y ) if R  = ∅ ; −1 (2) IntR (B1 ) ⊆ IntR (B2 ) if B1 ⊆ B2 ⊆ Y and Int−1 R (A2 ) ⊆ IntR (A1 ) if A1 ⊆ A2 ⊆ X . Remark 22 Conversely, it can be shown that, for any such relation Int on P(Y ) to P(X ), there exists a nonvoid relator R on X to Y such that Int = IntR [191]. Theorem 34 We have (1) ClR =



Cl R ;

(2) IntR =

R∈R



Int R .

R∈R

Corollary 4 The mapping (1) R → IntR is union-preserving;

(2) R → ClR is intersection-preserving.

Definition 2 For any x ∈ X and B ⊆ Y , we write (1) x ∈ clR (B) if {x} ∈ ClR (B) ;

(2) x ∈ intR (B) if {x} ∈ IntR (B).

The relations clR and intR are called the topological closure and topological interior generated by R, respectively. Theorem 35 For any x ∈ X and B ⊆ Y , we have (1) x ∈ intR (B) if and only if R (x) ⊆ B for some R ∈ R ; (2) x ∈ clR (B) if and only if R (x) ∩ B = ∅ for all R ∈ R. Theorem 36 For any A ⊆ X and B ⊆ Y , (1) A ∈ IntR (B) implies A ⊆ intR (B) ; (2) A ∩ clR (B) = ∅ implies A ∈ ClR (B). Remark 23 Later, we shall see that if R is nonvoid and topologically fine, then the converse implications are also true. Therefore, in this particular case the relations ClR and IntR are not stronger tools than clR and intR . ( For such a situation, see also [210].)

32

Á. Száz

Theorem 37 For any B ⊆ Y , we have (1) clR (B) = X \intR ( Y \B);

(2) intR (B) = X \clR ( Y \B).

Remark 24 By using appropriate complementations, assertion (1) can be written in the more concise forms c  (1) clR = intR ◦ CY ;

c  (2) clR = intR ◦ CY .

Theorem 38 We have (1) clR (∅) = ∅ if R = ∅;

(2) clR (B1 ) ⊆ clR (B2 ) if

B1 ⊆ B2 ⊆ Y .

Theorem 39 We have (1) intR (X ) = X if R = ∅;

(2) intR (B1 ) ⊆ intR (B2 ) if

B1 ⊆ B2 ⊆ Y .

Remark 25 Conversely, it can be shown that, for any such relation int on P (Y ) to X , there exists a nonvoid relator R on X to Y such that int = intR [191]. Theorem 40 We have (1) clR =

R∈R

(2) intR =

cl R ;

 R∈R

int R .

Corollary 5 The mapping (1) R → intR is union-preserving;

(2) R → clR is intersection-preserving.

Theorem 41 For any B ⊆ Y , we have (1) clR (B) =

R∈R

R −1 [ B ] ;

(2) intR (B) =

 R∈R

R −1 [ B c ]c .

Theorem 42 For any R ∈ R, A ⊆ X and B ⊆ Y , we have A ⊆ int R (B)

⇐⇒

cl R −1 (A) ⊆ B.

Remark 26 This theorem shows that the mappings A → clR −1 (A) and B → intR (B), where A ⊆ X and B ⊆ Y , also establish a Galois connection between P(X ) and P(Y ). This important closure-interior Galois connection, introduced first in [220], is not independent from the more familiar upper and lower bound Galois connection considered in our former paper [208]. It can, for instance, be used to briefly prove that, for any R ∈ R, A ⊆ X and B ⊆ Y , we have   (1) R −1 R [ A ]c ⊆ Ac ;

c  (2) B ⊆ R R −1 [ B ]c .

2 Contra Continuity Properties of Relations in Relator Spaces

33

Definition 3 For any B ⊆ Y , we write (1) B ∈ ER if intR (B) = ∅;

(2) B ∈ DR if clR (B) = X .

The members of the families ER and DR are called the fat sets and dense sests generated by R, respectively. Remark 27 The families ER and DR were first explicitly introduced by the present author in [189, 191]. Theorem 43 For any B ⊆ Y , we have (1) B ∈ ER if and only if R (x) ⊆ B for some x ∈ X and R ∈ R ; (2) B ∈ DR if and only if R (x) ∩ B = ∅ for all x ∈ X and R ∈ R. Theorem 44 For any B ⊆ Y , we have (1) B ∈ DR if and only if X = R −1 [ B ] for all R ∈ R ; (2) B ∈ ER if and only if X = R −1 [ B c ] for some R ∈ R. Theorem 45 For any B ⊆ Y , we have (1) B ∈ DR ⇐⇒

Bc ∈ / ER ;

(2) B ∈ ER ⇐⇒

Bc ∈ / DR .

Theorem 46 For any B ⊆ Y , we have (1) B ∈ DR if and only if B ∩ E = ∅ for all E ∈ ER ; (2) B ∈ ER if and only if B ∩ D = ∅ for all D ∈ DR . Theorem 47 We have (1) ∅ ∈ / DR if X = ∅ and R = ∅ ; (2) B ∈ DR and B ⊆ C ⊆ Y imply C ∈ DR . Theorem 48 We have (1) Y ∈ ER if X = ∅ and R = ∅ ; (2) B ∈ ER and B ⊆ C ⊆ Y imply C ∈ ER . Remark 28 Conversely, it can be shown that if A is a nonvoid, ascending family of subsets of a nonvoid set X , then there exists a nonvoid, preorder relator R on X such that A = ER [206]. Theorem 49 We have (1) ER =

 R∈R

ER ;

(2) DR =

R∈R

DR .

34

Á. Száz

Corollary 6 The mapping (1) R → ER is union-preserving;

(2) R → DR is intersection-preserving.

Remark 29 In addition to the above results, it is also worth mentioning that if in particular R is E –simple, then the stack (ascending family) ER has a base B with card (B) ≤ card (X ). ( For a proof, see Pataki [157].)

2.7 Some Further Structures Derived from Relators Notation 2 In this section, we shall already assume that R is a relator on X . Definition 4 For any A ⊆ X , we write (1) A ∈ τR if A ∈ IntR (A);

(2) A ∈ τ-R if Ac ∈ / ClR (A).

The members of the families τR and τ-R are called the proximally open sets and proximally closed sets generated by R, respectively. Remark 30 The families τR and τ-R were first explicitly introduced by the present author in [191]. In particular, the practical notation τ-R has been suggested by my former PhD student János Kurdics. Theorem 50 For any A ⊆ X , we have (1) A ∈ τR if and only if R [ A ] ⊆ A for some R ∈ R ; (2) A ∈ τ-R if and only if A ∩ R [ Ac ] = ∅ for some R ∈ R. Theorem 51 For any A ⊆ X , we have (1) A ∈ τ-R ⇐⇒

Ac ∈ τR ;

(2) A ∈ τR ⇐⇒

Ac ∈ τ-R .

Theorem 52 We have (1) τ-R = τR −1 ;

(2) τR = τ-R −1 .

Theorem 53 If R = ∅, then (1) { ∅, X } ⊆ τR ;

(2) { ∅, X } ⊆ τ-R .

Remark 31 Conversely, it can be shown that if A is a family of subsets of X containing ∅ and X , then there exists a nonvoid, preorder relator R on X such that A = τR [206].

2 Contra Continuity Properties of Relations in Relator Spaces

Theorem 54 We have  (1) τR = R∈R τ R ;

(2) τ-R =

35

 R∈R

τ- R .

Corollary 7 The mappings R → τR and R → τ-R are union-preserving. Definition 5 For any A ⊆ X , we write (1) A ∈ TR if A ⊆ intR (A)

(2) A ∈ FR if clR (A) ⊆ A.

The members of the families TR and FR are called the topologically open sets and topologically closed sets generated by R, respectively. Theorem 55 For any A ⊆ X , we have (1) A ∈ TR if and only if for each x ∈ A there exists R ∈ R such that R (x) ⊆ A ; (2) A ∈ FR if and only if for each x ∈ Ac there exists R ∈ R such that A ∩ R (x) = ∅. Theorem 56 For any A ⊆ X , we have (1) A ∈ FR ⇐⇒

A c ∈ TR ;

(2) A ∈ TR ⇐⇒

A c ∈ FR .

Theorem 57 We have (1) τR ⊆ TR ;

(2) τ-R ⊆ FR .

Theorem 58 For any R ∈ R, we have (1) τ R = T R ;

(2) τ- R = F R .

Theorem 59 The following assertions hold: (1) if A ∈ DR then for all V ∈ FR \{X } we have A\V = ∅ ; (2) if A ⊆ X such that there exists U ∈ TR \{∅} with U ⊆ A, then A ∈ ER . Theorem 60 We have (1) ∅ ∈ FR if R = ∅ ;

(2) A ⊆ FR implies



A ∈ FR .

Theorem 61 We have (1) X ∈ TR if R = ∅ ;

(2) A ⊆ TR implies



A ∈ TR .

Remark 32 Conversely, it can be shown that if A is a family of subsets of X such that X ∈ A and A is closed under arbitrary unions, then there exists a nonvoid, preorder relator R on X such that A = TR [206]. Therefore, analogously to proximities, closures and filters, topologies and their immediate generalizations should not also be studied without generalized uniformities.

36

Á. Száz

Theorem 62 We have (1)

 R∈R

T R ⊆ TR ;

(2)

 R∈R

F R ⊆ FR .

The fact that the corresponding equalities need not be true can be easily shown by using the following. Example 1 For any set X , with card (X ) > 2, there exists an equivalence relator R = { R1 , R2 } on X such that TR = T R1 ∪ T R2 . Namely, if x1 ∈ X and x2 ∈ X \{ x1 }, then by defining  2 Ri = { xi }2 ∪ X \{ xi }   for all i = 1, 2 we can see that { x1 , x2 } ∈ TR \ T R1 ∪ T R2 . Remark 33 In the light of the above disadvantage of the topologically open sets, it seems to be rather curious that analysis has been mainly based on topologically open sets. Open sets, as the most basic term in topology, were first suggested by Tietze (1923). They were later standardized by the fundamental books of Bourbaki (1940) and Kelley (1955). ( See Thron [228, p. 18].) Hausdorff (1914), Kuratowski (1922), Weil (1937), Tukey (1940), Efremoviˇc (1952), Császár (1960), Doiˇcinov (1964) and several other mathematicians, offered more convenient tools such as neighbourhoods, closures, uniformities, covers, proximities and convergences, for instance. Remark 34 The most powerful tools in a relator space (X, Y )(R) are the convergence LimR and adherence AdhR of a net of points or sets to another one. These relations, which should also be considered here, can be most naturally defined by using fat and dense sets. Namely, if ≤ is a certain order relation on a set X , then E≤ and D≤ are just the families of all residual and cofinal subsets of the ordered set X ( ≤ ), respectively. Note that now T≤ is just the family of all ascending subsets of X ( ≤ ). To clarify the advantage of fat sets over the open ones, it is also worth mentioning that if in particular X = R and R is a relation on X such that R(x) = { x − 1 } ∪ [ x, +∞ [ for all x ∈ X , then T R = { ∅, X }, but E R is quite large family. Namely, the supersets of each R(x), with x ∈ X , are also in E R .

2 Contra Continuity Properties of Relations in Relator Spaces

37

2.8 Some Important Closure Operations for Relators Notation 3 In this section, we shall assume that R is a relator on X to Y . Definition 6 The relators R∗ = R# = R∧ = R =



  

S ⊆ X ×Y :

∃ R∈R:

 R⊆S ;

S ⊆ X ×Y :

∀ A⊆X:

∃ R∈R:

S ⊆ X ×Y :

∀ x∈X:

∃ R∈R:

S ⊆ X ×Y :

 R[ A] ⊆ S[ A] ;  R (x) ⊆ S (x) ;

∀ x ∈X : ∃ u∈X : ∃ R∈R:

 R (u) ⊆ S (x)

are called the uniform, proximal, topological and paratopological closures (refinements) of the relator R, respectively. Thus, by the corresponding definitions, we can at once state the following. Theorem 63 We have R = R∧ = R = #

  

S ⊆ X ×Y :

∀ x∈X:

S ⊆ X ×Y :

∀ x∈X:

S ⊆ X ×Y :

∀ A⊆X:

 S(x) ∈ ER ;   x ∈ intR S(x) ;   A ∈ IntR S [ A ] .

In our former paper [225], by using Theorems 34, 40 and 54 and a particular theory of Pataki connections, we have proved the statements of the following two theorems in a unified way. Theorem 64 ∗, #, ∧ and  are closure operations for relators such that R ⊆ R∗ ⊆ R# ⊆ R∧ ⊆ R Hint By Definition 6, it is clear that these operations are extensive and increasing. Moreover, we can also easily see that the operation ∗ is idempotent, and the above inclusions hold. However, to prove directly that the operations #, ∧ and  are also idempotent may be an exhausting work. Corollary 8 We have  #  ∗ (1) R # = R ∗ = R # ;  ∧  ♦ with ♦ = ∗ or # ; (2) R ∧ = R ♦ = R ∧  ♦    ♦  (3) R = R = R with ♦ = ∗, # or ∧.

38

Á. Száz

Hint To prove (1), note that, by Theorem 64, we have  ∗  # R# ⊆ R# ⊆ R# = R#

and

 #  # R# ⊆ R∗ ⊆ R# = R#.

Thus, the corresponding equalities are also true. Theorem 65 For any relator S on X to Y , we have (1) S ⊆ R  ⇐⇒ S  ⊆ R  ⇐⇒ ES ⊆ ER ⇐⇒ DR ⊆ DS ; (2) S ⊆ R ∧ ⇐⇒ S ∧ ⊆ R ∧ ⇐⇒ intS ⊆ intR ⇐⇒ clR ⊆ clS ; (3) S ⊆ R # ⇐⇒ S # ⊆ R # ⇐⇒ IntS ⊆ IntR ⇐⇒ ClR ⊆ ClS . Corollary 9 The following assertions are true : (1) S = R  is the largest relator on X to Y such that ES = ER , or equivalently DS = D R ; (2) S = R ∧ is the largest relator on X to Y such that intS = intR , or equivalently clS = clR ; (3) S = R # is the largest relator on X to Y such that IntS = IntR , or equivalently ClS = ClR . Remark 35 To prove similar results for the operation ∗, we should use the relations LimR and AdhR . In some former papers, by using the corresponding definitions, we have also proved the following two theorems. Theorem 66 If R is nonvoid, then for any B ⊆ Y , we have   (1) IntR ∧ (B) = P intR (B) ;

 c (2) ClR ∧ (B) = P clR (B)c .

  Hint To prove the less obvious part of (1), note that if A ∈ P intR (B) , i. e., A ⊆ intR (B), then for each x ∈ A there exists Rx ∈ R such that Rx (x) ⊆ B. Hence, by defining S(x) = Rx (x) if x ∈ A, and S (x) = Y if x ∈ Ac , we can see that S ∈ R ∧ and S [ A ] ⊆ B. Therefore, A ∈ IntR ∧ (B) also holds. Remark 36 Note that  if R isnot supposed to be nonvoid, then instead of (1) we can only state that P intR (B) = IntR ∧ (B) ∪ {∅}. Moreover, in contrast to Theorem 36, for any A ⊆ X , we have A ∈ ClR ∧ (B) if and only if A ∩ clR (B) = ∅ Theorem 67 If R is nonvoid, then for any B ⊆ Y , we have (1) IntR  (B) = {∅} if B ∈ / ER and IntR  (B) = P (X ) if B ∈ ER ; (2) ClR  (B) = ∅ if B ∈ / DR and ClR  (B) = P (X )\{∅} if B ∈ DR .

2 Contra Continuity Properties of Relations in Relator Spaces

39

Proof If A ∈ IntR  (B), then there exists S ∈ R  such that S[ A ] ⊆ B. Therefore, if A = ∅, then there exists x ∈ X such that S(x) ⊆ B. Hence, by using that S (x) ∈ ER and ER is ascending, we can infer that B ∈ ER . Therefore, if B ∈ / ER , then we necessarily have IntR  (B) ⊆ {∅}. Moreover, since R = ∅, we can also note that R  = ∅, and thus ∅ ∈ IntR  (B). Therefore, the first part of assertion (1) is true. On the other hand, if B ∈ ER , then by defining R = X × B and using Theorem 8.3, we can see that R ∈ R  . Moreover, we can also note that R [ A ] ⊆ B, and thus A ∈ IntR  (B) for all A ⊆ X . Therefore, the second part of assertion (1) is also true. Now, assertion (2) can, in principle, be easily derived from assertion (1) by using Theorems 30 and 45. Corollary 10 If R is nonvoid, then for any B ⊆ Y , we have (1) clR  (B) = ∅ if B ∈ / DR and clR  (B) = X if B ∈ DR ; (2) intR  (B) = ∅ if B ∈ / ER and intR  (B) = X if B ∈ ER .

2.9 Some Further Important Unary Operations for Relators Notation 4 In this section, we shall already assume that R is a relator on X . The importance of the operation ∞ is also apparent from the following. Theorem 68 The following assertions are true : (1) ∞ is a closure operation for relations; (2) for any R, S ∈ R, we have S ⊆ R ∞ ⇐⇒ S ∞ ⊆ R ∞ ⇐⇒ τ R ⊆ τ S ⇐⇒ τ- R ⊆ τ- S ; (3) for any R ∈ R, S = R ∞ is the largest relation on X such that τ S = τ R , or equivalently τ- S = τ- R . Hint If x ∈ X , then by Theorem 10 it is clear that   R [ R ∞ (x) ] ⊆ R ∞ [ R ∞ (x) ] = R ∞ ◦ R ∞ (x) ⊆ R ∞ (x). Hence, by using Theorem 50, we can infer that R ∞ (x) ∈ τR . Now, if τ R ⊆ τ S holds, then we can see that R ∞ (x) ∈ τ S , and thus S [ R ∞ (x) ] ⊆ R ∞ (x). Hence, since x ∈ R ∞ (x), we can already infer that S(x) ⊆ R ∞ (x). Therefore, S ⊆ R ∞ also holds. This proves that τ R ⊆ τ S implies S ⊆ R ∞ . To prove the converse implication, note that if A ∈ τ R , then R [ A ] ⊆ A. Hence, by induction, it is clear that R n [ A ] ⊆ A for all n ∈ N. Now, since R 0 [ A ] =  X [ A ] = A, we can already see that

40

Á. Száz

R∞[ A ] =

 ∞

 ∞ ∞   Rn [ A ] = Rn [ A ] ⊆ A = A.

n=0

n=0

n=0

Therefore, if S ⊆ R ∞ holds, then S [A ] ⊆ R ∞ [ A ] ⊆ A, and thus A ∈ τ S also holds. Remark 37 A preliminary form of this theorem, and the fact that R ∞ (x) = for all x ∈ X , and thus R ∞ =





A ∈ τR : x ∈ A



{ R A : A ∈ τ R }, were first proved by Mala [123].

Now, as an immediate consequence of Theorems 54 and 68 and Corollary 9, we can also state Theorem 69 We have (1) τR = τR ∞ = τR # ;

(2) τ-R = τ-R ∞ = τ-R # ;

(3) τR = τR ∞# = τR #∞ ;

(4) τ-R = τ-R ∞# = τ-R #∞ .

Remark 38 Concerning the operation ∞, we can also easily prove that (1) R ∞ ⊆ R ∗ ∞ ⊆ R ∞∗ ⊆ R ∗ ; (2) R ∗∞ = R ∞∗ ∞ ; (3) R ∞∗ = R ∗ ∞∗ . However, it is now more important to note that, by using the arguments of Pataki [158] and Mala [123], we can also prove the following two theorems. Theorem 70 The following assertions are true : (1) # ∂ is a closure operation for relators ; (2) for any relator S on X , we have S ⊆ R #∂ ⇐⇒ S #∂ ⊆ R #∂ ⇐⇒ τS ⊆ τR ⇐⇒ τ-S ⊆ τ-R ; (3) S = R #∂ is the largest relator on X such that τS = τR , or equivalently τ-S = τ-R . Theorem 71 The following assertions are true : (1) # ∞ is a projection operation for relators ; (2) for any relator S on X , we have S ∞ ⊆ R # ⇐⇒ S #∞ ⊆ R #∞ ⇐⇒ τS ⊆ τR ⇐⇒ τ-S ⊆ τ-R ; (3) S = R #∞ is the largest preorder relator on X such that τS = τR , or equivalently τ-S = τ-R .

2 Contra Continuity Properties of Relations in Relator Spaces

41

Remark 39 In this respect, it is worth mentioning that the following assertions are also equivalent : (1) S ∞ ⊆ R # ;

(2) S #∞ ⊆ R # ;

(3) S ∞# ⊆ R ∞# .

The advantage of the projection operations # ∞ and ∞ # over the closure operation # ∂ lies mainly in the fact that, in contrast to # ∂, they are stable in the sense that they leave the relator { X 2 } fixed. Since, the structures T and F are not union-preserving, instead of an analogue of Theorem 70 we can only prove the following. Theorem 72 The following assertions are true : (1) ∧ ∂ is a preclosure operation for relators ; (2) for any relator S on X , we have TS ⊆ TR ⇐⇒ FS ⊆ FR =⇒ S ∧ ⊆ R ∧ ∂

=⇒ S ∧ ∂ ⊆ R ∧ ∂ .

  Remark 40 If card(X ) > 2, then by using the equivalence relator R = X 2 , considered first by Mala [123, Example 5.3], it can be shown that the operation ∧ ∂ is not idempotent [158, Example 7.2]. Now, by Theorems 65 and 66, we can also state the following two theorems. Theorem 73 We have (1) TR = TR ∧ ;

(2) TR = TR ∧ .

Theorem 74 If R is nonvoid, then (1) τR ∧ = TR ;

(2) τ-R ∧ = FR .

Hence, by Theorems 54 and 58, it is clear that we also have Corollary 11 If R is a nonvoid, then (1) TR =

 R∈R ∧

TR ;

(2) FR =

 R∈R ∧

FR .

Remark 41 Note that if in particular R = ∅, then by the definition of TR we have TR = {∅}. Moreover, if  in addition X = ∅, then by the definition of R ∧ we also have ∧ R = ∅. Thus, R∈R ∧ T R = ∅. Therefore, if R = ∅, but X = ∅, then the equalities stated in Corollary 11, and thus also in Theorems 74 and 66, do not hold. Moreover, from Theorem 71, by using Theorem 74 and Corollary 8, we can easily derive the following.

42

Á. Száz

Theorem 75 The following assertions are true : (1) ∧ ∞ is a modification operation for relators ; (2) if R is nonvoid, then for any nonvoid relator S on X , we have S ∧ ∞ ⊆ R ∧ ⇐⇒ S ∧∞ ⊆ R ∧∞ ⇐⇒ TS ⊆ TR ⇐⇒ FS ⊆ FR ; (3) if R is nonvoid, then S = R ∧∞ is the largest preorder relator on X such that TS = TR , or equivalently FS = FR . Remark 42 If card (X ) > 2 and R[ X 2 }, then by [123, Example 5.3] there does not exist a largest relator S on X such that TS = TR . ( For some other inconveniences, se also [158, Example 7.2] and [204, Example 10.11].) In the light of the several disadvantages of the topologically open sets and the results of Pervin [160], Fletcher and Lindgren [85] and the present author [206], it is rather curious that generalized topologies and minimal structures are still intensively investigated, without generalized uniformities, by a great number of mathematicians.

2.10 Proximal Interior and Closure Reversing Relations Notation 5 In this section, we shall assume that : (1) (X, Y )(R) and (Z , W )(S ) are relator spaces ; (2) F is a relation on X to Z and G is a relation on Y to W . Remark 43 To keep in mind the above assumptions, for any R ∈ R and S ∈ S , one can use the following diagram. F

X −−−−→ ⏐ ⏐ R

Z ⏐ ⏐

S

G

Y −−−−→ W Definition 7 The ordered pair (F, G) will be called (1) proximal closure reversing if for all A ⊆ X and B ⊆ Y A ∈ ClR (B) =⇒

  F [ A ] ∈ IntS G [ B ] ;

(2) proximal interior reversing if for all ∅ = A ⊆ X and B ⊆ Y A ∈ IntR (B) =⇒

  F [ A ] ∈ ClS G [ B ] .

2 Contra Continuity Properties of Relations in Relator Spaces

43

Remark 44 To see the necessity of the assumption A = ∅ in (2), note that if both R and S are nonvoid, then for any B ⊆ Y we have ∅ ∈ IntR (B) and F [ ∅ ] =  ∅ ∈ / ClS G [ B ] . Concerning property (2), we can also easily establish the following Theorem 76 The following assertions are equivalent : (1) (F, G) is proximal interior reversing ; (2) for all x ∈ X and B ⊆ Y x ∈ intR (B) =⇒

  F (x) ∈ ClS G [ B ] .

Proof If (1) holds, and x ∈ X and B ⊆ Y , then by taking A = {x} in Definition 7, we can see that {x} ∈ IntR (B) implies F [ {x} ] ∈ ClS (G [ B ] ). Hence, since F (x) = F [ {x} ], and x ∈ intR (B) is equivalent to {x} ∈ IntR (B), it is clear that (2) also holds. To prove the converse implication, suppose now that (2) holds, and A ⊆ X and B ⊆ Y such that A = ∅ and A ∈ IntR (B). Then, there exists x ∈ X such that x ∈ A. Moreover, since {x} ⊆ A, we also have {x} ∈ Int R (B), and thus x ∈ intR (B). Hence, by using (2), we can infer that F (x) ∈ ClS G [ B ] . Now, since F (x) = F [ {x} ] ⊆ F [ A ], we can already see that F [ A ] ∈ ClS G [ B ] , and thus (1) also holds. Now, by using this theorem, we can also prove the following two theorems. Theorem 77 The following assertions are equivalent : (1) (F, G) is proximal interior reversing ; (2) for all x ∈ X , R ∈ R and S ∈ S we have     S F (x) ∩ G R (x) = ∅ ; (3) for all x ∈ X , R ∈ R and S ∈ S there exist w ∈ W , z ∈ F (x) and y ∈ R (x) such that w ∈ S (z) and w ∈ G (y). Proof If x ∈ X  and R ∈ R, then because of R (x) ⊆ R (x) we evidently have x ∈ intR R (x) . Hence, if (1) holds, then by using Theorem 76 we can already infer that F (x) ∈ ClS ( G [ R (x) ]). Therefore, for any S ∈ S , we have S [ F (x) ] ∩ G [ R (x) ] = ∅, and thus assertion (2) also holds. On the other hand, if x ∈ X and B ⊆ Y such that x ∈ intR (B), then there exists R ∈ R such that R (x) ⊆ B. This implies that G [ R (x) ] ⊆ G [ B ]. Moreover, if (2) holds, then for any S ∈ S we have S [ F (x) ] ∩ G [ R (x) ] = ∅. Therefore, for any S ∈ S ,  we also  have S [ F (x) ] ∩ G [ B ] = ∅. Hence, we already see that F (x) ∈ ClS G [ B ] . Thus, by Theorem 76, assertion (1) also holds. Now, to complete the proof, it remains to note only that (3) is a detailed reformulation of (2).

44

Á. Száz

Remark 45 Note that assertion (2) can also be reformulated in the form that, for all ∅ = A ⊆ X , R ∈ R and S ∈ S ,     S F [ A ] ∩ G R [ A ] = ∅. Theorem 78 The following assertions are equivalent : (1) (F, G) is proximal interior reversing ; (2) for all x ∈ X and B ⊆ Y F (x) ∈ IntS



G [ B ]c



  =⇒ x ∈ clR B c ;

(3) for all x ∈ X and D ⊆ W    F (x) ∈ IntS D) =⇒ x ∈ clR G −1 [ D ] . Proof By using Theorems 30 and 37, we can see that, for any x ∈ X and B ⊆ Y , the following implications are equivalent :   x ∈ intR (B) =⇒ F (x) ∈ ClS G [ B ] ,   F (x) ∈ / ClS G [ B ] =⇒ x ∈ / intR (B),     c F (x) ∈ IntS G [ B ] =⇒ x ∈ clR B c . Therefore, by Theorem 76, assertions (1) and (2) are equivalent. On the other hand, if assertion (2) holds, and x ∈ X and D ⊆ W , then by using the inclusion c  D ⊆ G G −1 [ D ] c derivable from Remark 26, and assertion (2), we can see that F (x) ∈ IntS (D) =⇒ F (x) ∈ IntS



  c  G G −1 [ D ]c =⇒ x ∈ clR G −1 [ D ] .

Thus, assertion (3) also holds. While, if assertion (3) holds, and x ∈ X and B ⊆ Y , then by using assertion (3) and the inclusion   G −1 G [ B ] c ⊆ B c derivable also from Remark 26, we can see that F (x) ∈ IntS



G [ B ]c



     =⇒ x ∈ clR G −1 G [ B ]c =⇒ x ∈ clR B c .

Thus, assertion (2) also holds.

2 Contra Continuity Properties of Relations in Relator Spaces

45

Remark 46 Note that assertion (3) is equivalent to the property that, for all ∅ = A ⊆ X and D ⊆ W ,  F [ A ] ∈ IntS D) =⇒

  A ∈ ClR G −1 [ D ] .

Now, by using Theorem 78 and Remark 46, we can also prove the following two particular theorems. Theorem 79 If in particular G is onto W and G −1 is a function, then the following assertions are equivalent : (1) (F, G) is proximal interior reversing ; (2) for all x ∈ X and B ⊆ Y   F (x) ∈ IntS ( G [ B ] ) =⇒ x ∈ clR B . Proof From Theorem 78, we know that (1) is equivalent to the assertion that (a) for all x ∈ X and B ⊆ Y F (x) ∈ IntS



G [ B ]c



  =⇒ x ∈ clR B c .

Moreover, from Remark 15, we know that G [ B ]c = G [ B c ]. Therefore, (a) is equivalent to the assertion that (b) for all x ∈ X and B ⊆ Y F (x) ∈ IntS



G [ Bc ]



  =⇒ x ∈ clR B c .

Hence, by taking B c in place of B, we can see that (b) is equivalent to (2). Therefore, (1) and (2) are also equivalent. Theorem 80 If f is a function on X onto Z such that ( f, G) is proximal interior reversing with respect to R and S , then ( f −1 , G −1 ) is proximal interior reversing with respect to S and R. Proof If ∅ = C ⊆ Z and D ⊆ W , then by using the inclusion f



 f −1 [ C ] ⊆ C

derivable from Theorem 14, and Remark 46, we can see that C ∈ IntS (D) =⇒ f



   f −1 [ C ] ∈ IntS (D) =⇒ f −1 [ C ] ∈ ClR G −1 [ D ] .

Thus, the required assertion is also true. However, instead of some close analogues of the above theorems, we can only prove the following theorems.

46

Á. Száz

Theorem 81 If (F, G) is proximal closure reversing, then for all x ∈ X and B⊆Y   x ∈ clR (B) =⇒ F (x) ∈ IntS G [ B ] . Theorem 82 The following assertions are equivalent : (1) (F, G) is proximal closure reversing ; (2) if A ⊆ X and B ⊆ Y such that R [ A ] ∩ B = ∅ for all R ∈ R, then there exists S ∈ S such that   S F [ A] ⊆ G [ B ]; (3) if A ⊆ X and B ⊆ Y such that R [ A ] ∩ B = ∅ for all R ∈ R, then there exists S ∈ S such that, for all u ∈ A and z ∈ F (u) and w ∈ S (z), there exists v ∈ B such that w ∈ G (v). Theorem 83 The following assertions are equivalent : (1) (F, G) is proximal closure reversing ; (2) for all A ⊆ X and B ⊆ Y F [ A ] ∈ ClS



G [ B ]c



=⇒

  A ∈ IntR B c ;

(3) for all A ⊆ X and D ⊆ W  F [ A ] ∈ ClS D) =⇒

A ∈ IntR



 G −1 [ D ] .

Theorem 84 If in particular G is onto W and G −1 is a function, then the following assertions are equivalent : (1) (F, G) is proximal closure reversing ; (2) for all A ⊆ X and B ⊆ Y F [ A ] ∈ ClS ( G [ B] ) =⇒

A ∈ IntR (B).

Theorem 85 If F is onto Z and (F, G) is proximal closure reversing with respect to R and S , then ( F −1 , G −1 ) is proximal closure reversing with respect to S and R. Proof If C ⊆ Z and D ⊆ W , then by using the inclusion   C ⊆ F F −1 [ C ] derivable from Theorem 13, and Theorem 83, we can see that     C ∈ ClS (D) =⇒ F F −1 [ C ] ∈ ClS (D) =⇒ F −1 [ C ] ∈ IntR G −1 [ D ] . Thus, the required assertion is also true.

2 Contra Continuity Properties of Relations in Relator Spaces

47

Remark 47 In the above results, because of Corollary 9, we may write R # and S # in place of R and S , respectively.

2.11 Topological Interior and Closure Reversing Relations Notation 6 In this section, somewhat more specially, we shall assume that (1) (X, Y )(R) and (Z , W )(S ) are relator spaces ; (2) f is a function of X to Z and G is a relation on Y to W . Remark 48 Now, to keep in mind the above assumptions, for any R ∈ R and S ∈ S , one can use the following diagram. f

X −−−−→ ⏐ ⏐ R

Z ⏐ ⏐

S

G

Y −−−−→ W Definition 8 The ordered pair ( f, G) will be called (1) topological closure reversing if for all x ∈ X and B ⊆ Y x ∈ clR (B) =⇒

  f (x) ∈ intS G [ B ] ;

(2) topological interior reversing if for all x ∈ X and B ⊆ Y x ∈ intR (B) =⇒

  f (x) ∈ clS G [ B ] .

Remark 49 Note that if X was not supposed to be the domain of f , then the above implications could be required to hold only for all x ∈ D f . Now, by Theorem 76, we can at once state the following Theorem 86 The following assertions are equivalent : (1) ( f, G) is proximal interior reversing ; (2) ( f, G) is topological interior reversing. Hence, by the corresponding results of Sect. 2.10, it is clear that we also have the following theorems. Theorem 87 The following assertions are equivalent : (1) ( f, G) is topological interior reversing ;

48

Á. Száz

(2) for all x ∈ X , R ∈ R and S ∈ S we have S



   f (x) ∩ G R (x) = ∅ ;

(3) for all x ∈ X , R ∈ R and S ∈ S there exist w ∈ W and y ∈ R (x) such that w ∈ S f (x) and w ∈ G (y). Remark 50 Assertion (2) can again be reformulated in the form that, for all ∅ = A ⊆ X , R ∈ R and S ∈ S ,     S f [ A ] ∩ G R [ A ] = ∅. Theorem 88 The following assertions are equivalent : (1) ( f, G) is topological interior reversing ; (2) for all x ∈ X and B ⊆ Y f (x) ∈ intS



G [ B ]c



  =⇒ x ∈ clR B c ;

(3) for all x ∈ X and D ⊆ W    f (x) ∈ intS D) =⇒ x ∈ clR G −1 [ D ] . Remark 51 Assertion (3) is again equivalent to the property that, for all ∅ = A ⊆ X and D ⊆ W ,  f [ A ] ∈ IntS D) =⇒

  A ∈ ClR G −1 [ D ] .

Theorem 89 If in particular G is onto W and G −1 is a function, then the following assertions are equivalent : (1) ( f, G) is proximal interior reversing ; (2) for all x ∈ X and B ⊆ Y   f (x) ∈ intS ( G [ B ] ) =⇒ x ∈ clR B . Theorem 90 If f is injective and onto Z , and ( f, G) is topological interior reversing with respect to R and S , then ( f −1 , G −1 ) is topological interior reversing with respect to S and R. Now, by Theorem 81, we can also state the following. Theorem 91 If ( f, G) is proximal closure reversing, then ( f, G) is, in particular, topological closure reversing. Moreover, analogously to the remaining results of Sect. 10, we can also easily prove the following theorems.

2 Contra Continuity Properties of Relations in Relator Spaces

49

Theorem 92 The following assertions are equivalent : (1) ( f, G) is topological closure reversing ; (2) if x ∈ X and B ⊆ Y such that R (x) ∩ B = ∅ for all R ∈ R, then there exists S ∈ S such that   S f (x) ⊆ G [ B ] ; (3) if x ∈ X and B ⊆ Y such that for all R ∈ R there exists  y ∈ B such that y ∈ R (x), then there exists S ∈ S such that for all w ∈ S f (x) there exists v ∈ B such that w ∈ G (v). Theorem 93 The following assertions are equivalent : (1) ( f, G) is topological closure reversing ; (2) for all x ∈ X and B ⊆ Y , f (x) ∈ clS



G [ B ]c



  =⇒ x ∈ intR B c ;

(3) for all x ∈ X and D ⊆ W ,    f (x) ∈ clS D) =⇒ x ∈ intR G −1 [ D ] . Theorem 94 If f is injective and onto Z , and ( f, G) is topological closure reversing with respect to R and S , then ( f −1 , G −1 ) is topological closure reversing with respect to S and R. Remark 52 In the above results, because of Corollary 9, we may write R ∧ and S ∧ in place of R and S , respectively.

2.12 Fatness and Denseness Reversing Relations Notation 7 In this section, we shall assume that (X, Y )(R) and (Z , W )(S ) are relator spaces, and G is a relation on Y to W . Remark 53 To keep in mind the above assumptions, for any R ∈ R and S ∈ S , one can use the following diagram. X ⏐ ⏐ R

Z ⏐ ⏐

S G

Y −−−−→ W Definition 9 The relation G will be called (1) fatness reversing if B ∈ ER

=⇒ G [ B ] ∈ DS ;

50

Á. Száz

(2) denseness reversing if B ∈ DR

=⇒ G [ B ] ∈ ES .

Theorem 95 If there exists a function f of X to Z such that ( f, G) is topological interior reversing, then   (1) B ∈ ER implies clS G [ B ] = ∅ ;   −1 (2) D ∈ ES implies clR G [ D ] = ∅ if Z = f [ X ]. Proof If B ∈ ER , then there exists x ∈ X such that  x ∈ intR (B). Hence, by using Definition 8, we can infer that f (x) ∈ clS G [ B ] . Therefore, assertion (1) is true. While, if D ∈ ES , then there exists z ∈ Z such that z ∈ intS (D). Moreover, since Z = f [ X ], there exists x ∈ X such that z = f (x). Therefore, we have f (x) ∈ intS (D). Hence, by using Theorem 88, we can already infer that x ∈ clR G −1 [ D ] . Therefore, assertion (2) is also true. Theorem 96 The following assertions are equivalent : (1) (2) (3) (4) (5) (6)

G ∈ DR S ; G is fatness reversing ; S −1 ◦ {G} ◦ R ⊆ {X × Z } ;    Z = S −1 G R(x) ] for all x ∈ X , R ∈ R and S ∈ S ; S (z) ∩ G [ R(x) ] = ∅ for all x ∈ X , z ∈ Z , R ∈ R and S ∈ S ; for all x ∈ X , z ∈ Z , R ∈ R and S ∈ S there exist y ∈ R(x) and w ∈ S(z) such that w ∈ G (y).

Proof If x ∈ X and R ∈ R, then R(x) ∈ ER . Hence, if assertion (2) holds, we can already  infer   that G [ R(x) ] ∈ DS . Therefore, by Theorem 44, we have Z = S −1 G R(x) for all S ∈ S . Thus, assertion (4) also holds. On the other hand, if B ∈ ER , then there exist x ∈ X and R ∈ R such that R(x) ⊆ B. This implies that     S −1 G [ R(x) ] ⊆ S −1 G [ B ]   for all S ∈ S . Hence, if assertion (4) holds, we can infer that Z = S −1 G [ B ] for all S ∈ S . Therefore, by Theorem 44, we have G [ B ] ∈ DS . Thus, assertion (2) also holds. Now, to complete the proof, it remains to note only that, by the corresponding definitions, assertions (3)–(6) are equivalent. Moreover, by Theorems 41 and 29, we have  cl RS (G) = R  S )−1 [ G ] = S −1 ◦ G ◦ R for all R ∈ R and S ∈ S . Therefore, assertions (1) and (3) are also equivalent.

2 Contra Continuity Properties of Relations in Relator Spaces

51

Corollary 12 The following assertions are equivalent : (1) G is fatness reversing ;  −1 (2) S  ◦ {G} ◦ R  ⊆ {X × Z }. Proof By Corollary 9, we have ER = ER  and DS = DS  . Hence, by Definition 9, we can see that the following assertions are equivalent : (a) G is fatness reversing with respect to R and S ; (b) G is fatness reversing with respect to R  and S  . Therefore, by Theorem 96, assertions (1) and (2) are also equivalent. Remark 54 Note that if in particular both R and S are nonvoid, then instead of the corresponding inclusions in Theorem 96 and Corollary 12 we may write equalities. Theorem 97 The following assertions are equivalent : (2) G −1 is fatness reversing.

(1) G is fatness reversing ; Proof To prove this, note that S −1 ◦ {G} ◦ R ⊆ {X × Z } ⇐⇒



S −1 ◦ {G} ◦ R

−1

⊆ {X × Z }−1

⇐⇒ R −1 ◦ { G −1 } ◦ S ⊆ {Z × X }. Therefore, by Theorem 96, assertions (1) and (2) are also equivalent. Theorem 98 If in particular G is onto W and G −1 is a function, then the following assertions are equivalent : (1) G is fatness reversing ; (2) G [ B ] ∈ ES implies B ∈ DR for all B ⊆ Y . Proof By Theorem 45 and Remark 15, it is clear that, for any B ⊆ Y , the following implications are equivalent : B ∈ ER =⇒ G [ B ] ∈ DS ; G[B]∈ / DS =⇒ B ∈ / ER ; G [ B ]c ∈ ES G [ B c ] ∈ ES

=⇒ B c ∈ DR ; =⇒ B c ∈ DR .

Hence, by taking B c in place of B, we can see that assertions (1) and (2) are also equivalent. Remark 55 More generally, we can state that (1) implies (2) if G is onto W , and (2) implies (1) if G −1 is a function. Now, analogously to Theorem 95, we can also easily prove the following. Theorem 99 If there exists a function f of X to Z such that ( f, G) is topological closure reversing, then

52

Á. Száz

  (1) B ∈ DR implies f [ X ] ⊆ intS G [ B ] ;  −1  (2) D ∈ DS implies X = intR G [ D ] . Proof If B ∈ DR , then for any x ∈ X we  have x ∈ clR (B). Hence, by using Definition 8, we can infer that f (x) ∈ intS G [ B ] . Therefore, assertion (1) is true. While, if D ∈ DS , then for any z ∈ Z we have z ∈ clS (D). Thus, in particular, for any x ∈ X, we have f (x) ∈ clS (D). Hence, by using Theorem 93, we can infer that x ∈ intR G −1 [ D ] . Therefore, assertion (2) is also true. Remark 56 Theorems 95 and 99 strongly suggest that, in addition to the families ER and DR , the families 

B ⊆ Y : clR (B) = ∅



and



B ⊆ Y : intR (B) = X



should have also been investigated in our former papers on relators. Now, by the X = Y , Z = W and f = G particular case of Theorem 99, we can also state the following. Corollary 13 If f is a topological closure reversing function of one relator space X (R) to another Y (S ), then (1) A ∈ DR implies f [ A ] ∈ TS ;

(2) B ∈ DS implies f −1 [ B ] ∈ TR .

While, instead of an analogue of Theorem 96, we can only prove the following. Theorem 100 The following assertions are equivalent : (1) G is denseness reversing, (2) if B ⊆ Y such that X = R −1 [ B ] for all R ∈ R, then there exist z ∈ Z and S ∈ S such that S (z) ⊆ G [ B ] ; (3) if B ⊆ Y such that for all x ∈ X and R ∈ R there exists y ∈ B such that y ∈ R (x), then there exist z ∈ Z and S ∈ S such that for all w ∈ S (z) there exists v ∈ B such that w ∈ G (v). However, in addition to Theorem 97, we can still prove the following. Theorem 101 The following assertions are equivalent : (1) G is denseness reversing ;

(2) G −1 is denseness reversing.

Proof Suppose that (1) holds and D ∈ DS . Then, for any B ∈ DR , we have G [ B ] ∈ ES . Hence, by using Theorem 46, we can infer that G [ B ] ∩ D = ∅. This implies that B ∩ G −1 [ D ] = ∅. Therefore, by Theorem 46, we have G −1 [ D ] ∈ ER . Thus, (2) also holds. Now, the converse implication (2) =⇒ (1) is quite obvious from the fact that −1  F = F −1 . Now, analogously to Theorem 98, we can also easily prove the following

2 Contra Continuity Properties of Relations in Relator Spaces

53

Theorem 102 If in particular G is onto W and G −1 is a function, then the following assertions are equivalent : (1) G is denseness reversing ; (2) G [ B ] ∈ DS implies B ∈ ER for all B ⊆ Y . Remark 57 More generally, we can state that (1) implies (2) if G is onto W , and (2) implies (1) if G −1 is a function.

2.13 Proximal Openness and Closedness Reversing Relations Notation 8 In this and the subsequent section, we shall assume that X (R) and Y (S ) are relator spaces, and F is a relation on X to Y . Remark 58 Now, to illustrate the above assumptions, for any R ∈ R and S ∈ S , one can use the following diagram. R

F

S

X −→ X −→ Y −→ Y Definition 10 The relation F will be called (1) proximal openness reversing if A ∈ τR =⇒

F [ A ] ∈ τ-S ;

(2) proximal closedness reversing if A ∈ τ-R =⇒

F [ A ] ∈ τS .

Remark 59 By Theorem 7.6, we have τ-S = τS −1 . Thus, implication (1) is equivalent to the implication A ∈ τR =⇒ F [ A ] ∈ τS −1 . This shows that the properties of the proximal openness reversing relations can be immediately derived from those of the proximal openness preserving ones. However, to keep the present section also as self-contained as possible, it seems now more convenient to apply some direct proofs. Theorem 103 The following assertions are equivalent :  # (1) F ◦ R ∞ ⊆ S −1 ◦ F ; (2) F is proximal openness reversing ; (3) for all A ⊆ X and R ∈ R there exists S ∈ S such that     S −1 F [ A ] ⊆ F R ∞ [ A ] ;

54

Á. Száz

(4) for all A ⊆ X and R ∈ R there exists S ∈ S such that for all x ∈ A, z ∈ F (x) and w ∈ S −1 (z) there exist u ∈ A and v ∈ R ∞ (u) such that w ∈ F (v). Proof If A ⊆ X and R ∈ R, then because of       R R∞ [ A ] ⊆ R∞ R∞ [ A ] = R∞ ◦ R∞ [ A ] = R∞ [ A ]   we have R ∞ [ A ] ∈ τR . Hence, if (2) holds, we can infer that F R ∞ [ A ] ∈ τ-S . By Theorem 52, this implies that F R ∞ [ A ] ∈ τS −1 . Thus, there exists S ∈ S such that      S −1 F R ∞ [ A ] ⊆ F R ∞ [ A ] . and thus A ⊆ R ∞ [ A ], we can already infer Hence, by R∞  using that   is∞reflexive,  −1 F [ A ] ⊆ F R [ A ] . Therefore, assertion (3) also holds. that S On the other hand, if A ∈ τR , then by Theorem 69 we have A ∈ τR ∞ . There ∞ [ A ] ⊆ fore, there exists R ∈ R such that R ∞ [ A ] ⊆ A. This implies that F R  −1 F [ A] ⊆ F [ A ]. Moreover, if (3) holds, then there exists S∈   S such that S  ∞ −1 F [ A ] ⊆ F [ A ]. Hence, we can infer F R [ A ] . Therefore, we also have S that F [ A ] ∈ τS −1 . By Theorem 52, this implies that F [ A ] ∈ τ-S . Therefore, assertion (2) also holds. Now, to complete the proof, it remains to note only that (1) is a concise reformulation of (3), and (4) is a detailed reformulation of (3). Remark 60 Assertion (4) can also be reformulated in a more detailed form, by using that (a) w ∈ S −1 (z) if and only if z ∈ S (w) ; n in X such that (b) v ∈ R ∞ (u) if and only if there exists a finite family ( xi )i=0 x0 = u, xn = v and xi ∈ R ( xi−1 ) for all i = 1, 2, . . . , n. However, it is now more important to note that we also have Corollary 14 The following assertions are equivalent : (1) F is proximal openness reversing ;

 # −1  # (2) F ◦ R # ∞ ⊆ S −1 ◦ F ; (3) F ◦ R ∞ ⊆ S∞ ◦ F . Proof To derive this statement from Theorem 103, note that by Theorem 69 we have τR = τR # and τ-S = τ-S ∞ . Therefore, the following assertions are equivalent : (a) F is proximal openness reversing with respect to R and S ; (b) F is proximal openness reversing with respect to R # and S ; (c) F is proximal openness reversing with respect to R and S ∞ . Remark 61 If in particular R is proximal in the sense that, for any A ⊆ X and R ∈ R, there exists U ∈ τR such that A ⊆ U ⊆ R [ A ], then by [225, Theorem #  62] we have R # = R ∞ .

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55

Therefore, by using Theorem 64 and [225, Theorem 51], we can see that #   #  # ⇐⇒ F ◦ R ∞ ⊆ S −1 ◦ F F ◦ R ∞ ⊆ S −1 ◦ F # #  −1 #    #  # ⇐⇒ F ◦ R ∞ ⊆ S ◦F ⇐⇒ F ◦ R # ⊆ S −1 ◦ F  #  #  # ⇐⇒ F ◦ R ⊆ S −1 ◦ F ⇐⇒ F ◦ R ⊆ S −1 ◦ F . Thus, instead of R ∞ we may write R in Theorem 103 and Corollary 14. Concerning proximal closedness reversing relations, it is convenient to prove first the following two theorems. Theorem 104 If in particular F is onto Y and F −1 is a function, then the following assertions are equivalent : (1) F is proximal openness reversing ; (2) F is proximal closedness reversing. Proof By using Theorem 51 and Remark 15, we can see that the following implications are equivalent : A ∈ τR =⇒ F [ A ] ∈ τ-S ; Ac ∈ τ-R =⇒ F [ A ]c ∈ τS ; Ac ∈ τ-R =⇒ F [ Ac ] ∈ τS . Hence, by taking Ac in place of A, we can see that assertions (1) and (2) are also equivalent. Theorem 105 The following assertions are equivalent : (1) F is proximal closedness reversing with respect to R and S ; (2) F is proximal openness reversing with respect to R −1 and S −1 . Proof By Theorem 52, we have τ-R = τR −1 and τS = τ-S −1 . Therefore, the implications (a) A ∈ τ-R =⇒ F [ A ] ∈ τS ,

(b) A ∈ τR −1 =⇒ F [ A ] ∈ τ-S −1

are equivalent. Thus, assertions (1) and (2) are also equivalent. Now, by using this theorem, from our former results on proximal openness reversing relations, we can immediately derive the following. Theorem 106 The following assertions are equivalent :  −1  # (1) F ◦ R ∞ ⊆ S ◦F ; (2) F is proximal closedness reversing ;

56

Á. Száz

(3) for all A ⊆ X and R ∈ R there exists S ∈ S such that     −1 S F [ A ] ⊆ F R∞ [ A ] ; (4) for all A ⊆ X and R ∈ R there exists S ∈ S such that for all x ∈ A,  −1 y ∈ F (x) and z ∈ S(y) there exist u ∈ A and v ∈ R ∞ (u) such that z ∈ F (v). Corollary 15 The following assertions are equivalent : (1) F is proximal closedness reversing ;  −1  −1  ∞ #  # (2) F ◦ R # ∞ ⊆ S ◦F ; (3) F ◦ R ∞ ⊆ S ◦F .

2.14 Topological Openness and Closedness Reversing Relations Definition 11 Under the assumptions of Notation 8, the relation F will be called (1) topological openness reversing if A ∈ TR =⇒

F [ A ] ∈ FS ;

(2) topological closedness reversing if A ∈ FR =⇒

F [ A ] ∈ TS .

Remark 62 In the sequel, in addition to the assumptions of Notation 8, we shall assume that both R and S are nonvoid. It may be a tiring work to determine that which of the subsequent statements remain true without these plausible requirements. Theorem 107 The following assertions are equivalent : (1) F is topological openness reversing with respect to R and S ; (2) F is proximal openness reversing with respect to R ∧ and S ∧ . Proof By Theorem 74, we have TR = τR ∧ and FS = τ-S ∧ . Therefore, the implications (a) A ∈ TR =⇒

F [ A ] ∈ FS ,

(b) A ∈ τR ∧ =⇒ F [ A ] ∈ τ-S ∧

are equivalent. Thus, by Definitions 11 and 10, assertions (1) and (2) are also equivalent. By using this theorem, from Theorem 103 and its corollary we can immediately derive the following theorem and corollary.

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Theorem 108 The following assertions are equivalent :  # (1) F ◦ R ∧ ∞ ⊆ S ∧−1 ◦ F ; (2) F is topological openness reversing ; (3) for all A ⊆ X and R ∈ R ∧ there exists S ∈ S ∧ such that     S −1 F [ A ] ⊆ F R ∞ [ A ] ; (4) for all A ⊆ X and R ∈ R ∧ there exists S ∈ S ∧ such that for all x ∈ A, z ∈ F (x) and w ∈ S −1 (z) there exist u ∈ A and v ∈ R ∞ (u) such that w ∈ F (v). Corollary 16 The following assertions are equivalent :

 # −1 S ∧∞ ◦ F ; (1) F ◦ R ∧ ∞ ⊆ (2) F is topological openness reversing. Remark 63 If in particular R is topological in the sense that, for any x ∈ X and R ∈ R, there exists U ∈ TR such that x ∈ U ⊆ R (x), then from [225, Theorem 64] we can see that R ∧ is proximal. Therefore, by Remark 61, instead of R ∧ ∞ we may write R ∧ in Theorem 108 and Corollary 16. Analogously to Theorem 107, we can also easily establish the following. Theorem 109 The following assertions are equivalent : (1) F is topological closedness reversing with respect to R and S ; (2) F is proximal closedness reversing with respect to R ∧ and S ∧ . By using this theorem, from Theorems 104 and 106 and Corollary 15, we can immediately derive the following two theorems and corollary. Theorem 110 If in particular F is onto Y and F −1 is a function, then the following assertions are equivalent : (1) F is topological openness reversing ; (2) F is topological closedness reversing. Remark 64 From the proof of Theorem 104, by Theorem 56, it is clear that this theorem does not actually need the assumptions made in Remark 62. Theorem 111 The following assertions are equivalent :  −1  ∧ # (1) F ◦ R ∧ ∞ ⊆ S ◦F ; (2) F is topological closedness reversing ;

58

Á. Száz

(3) for all A ⊆ X and R ∈ R ∧ there exists S ∈ S ∧ such that     −1 S F [ A ] ⊆ F R∞ [ A ] ; (4) for all A ⊆ X and R ∈ R ∧ there exists S ∈ S ∧ such that for all x ∈ A,  −1 y ∈ F (x) and z ∈ S(y) there exist u ∈ A and v ∈ R ∞ (u) such that z ∈ F (v). Corollary 17 The following assertions are equivalent :  # −1  ∧∞ (1) F ◦ R ∧ ∞ ⊆ S ◦F ; (2) F is topological closedness reversing. −1  Remark 65 By using Theorem 11 and the plausible notation R ∨ = R ∧ , the  # above inclusion can be written in the form that F ◦ R ∨∞ ⊆ S ∧∞ ◦ F . Moreover, by using the inversion-compatibility of the operation #, assertion (1) can also be reformulated in the form that #  R ∧ ∞ ◦ F −1 ⊆ F −1 ◦ S ∨∞ . This shows that the topological closedness reversingness of F is a certain mixedtype upper continuity of F −1 . ( See [225, Definition 2] and [218, Definition 4.8].)

2.15 Contra Continuity Properties of the Identity Function Notation 9 In this section, we shall assume that R and S are relators on the same set X . Moreover, we shall consider the identity function  X of X as a function of X (R) to X (S ). Thus, by specializing our former results to  X , we can easily establish the following theorems. Theorem 112 The following assertions are equivalent : (1) (2) (3) (4) (5)

intR ⊆ clS ; S −1 ◦ R is reflexive on X ;  X is proximal interior reversing ;  X is topological interior reversing ; ∅ = A ∈ IntR (B) implies A ∈ ClS (B).

Proof To check the equivalence of (2) and (4), note that  X [ A ] = A for all A ⊆ X . Therefore, by Theorem 87, assertion (3) is equivalent to the statement that, for all x ∈ X and R ∈ R and S ∈ S , we have

2 Contra Continuity Properties of Relations in Relator Spaces

S (x) ∩ R(x) = ∅,

i. e.,

59

x ∈ S −1 [ R (x) ].

That is, the relation S −1 ◦ R is reflexive on X for all R ∈ R and S ∈ S . Remark 66 To give a direct proof for the implication (1) =⇒ (5), one can note that if A ∈ IntR (B) and x ∈ A, then x ∈ intR (B). Hence, if (1) holds, we can infer that x ∈ clS (B). This implies that A ∈ ClS (B). Corollary 18 If both R and S are reflexive on X , then  X is both proximal and topological interior reversing. Proof Namely, in this case S −1 ◦ R is also reflexive on X . Thus, Theorem 112 can be applied to get the required assertion. Theorem 113 The following assertions are equivalent : (2)  X is proximal closure reversing ; (1) ClR ⊆ IntS ; (3) if A, B ⊆ X such that R [ A ] ∩ B = ∅ for all R ∈ R, then there exists S ∈ S such that S [ A ] ⊆ B. Theorem 114 The following assertions are equivalent : (1) clR ⊆ intS ; (2)  X is topological closure reversing ; (3) if x ∈ X and A ⊆ X such that R (x) ∩ A = ∅ for all R ∈ R, then there exists S ∈ S such that S (x) ⊆ A. Corollary 19 If there exists x ∈ X such that x ∈ R (x) for all R ∈ R and S (x)\{x} = ∅ for all S ∈ S , then  X is neither topological nor proximal closure reversing. Now, by Theorems 95 and 99, we can also state the following. Theorem 115 The following assertions hold : (1) if  X is topological interior reversing, then A ∈ ER =⇒ clS (A) = ∅ ; (2) if  X is topological closure reversing, then B ∈ DS

=⇒ X = intR (B).

Corollary 20 If  X is topological closure reversing, then DS ⊆ TR . Moreover, by Theorem 96 and Definition 9, we can also state the following two theorems. Theorem 116 The following assertions are equivalent :

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Á. Száz

(1) ER ⊆ DS ;

(2)  X ∈ DR S ;

  (3) S −1 ◦ R ⊆ X 2 ;

(4)  X is fatness reversing.

Remark 67 Note that if in particular   both R and S are nonvoid, then instead of (3) we may write S −1 ◦ R = X 2 . Theorem 117 The following assertions are equivalent : (1) DR ⊆ ES ;

(2)  X is denseness reversing.

Remark 68 Note that in the above theorems, by Theorems 80, 85, 90, 94, 97 and 101, we may write S and R in place of R and S , respectively. Finally, we note that, by specializing the corresponding results of Sects. 2.13 and 2.14, we can also at once state the following two theorems. Theorem 118 The following assertions are equivalent : #  (1) τR ⊆ τ-S ; (2) R ∞ ⊆ S −1 ; (3)  X is proximal openness reversing ; (4)  X is proximal closedness reversing ; (5) for all A ⊆ X and R ∈ R there exists S ∈ S such that S −1 [ A ] ⊆ R ∞ [ A ]. Remark 69 Note that, because of the inversion compatibilities of the projection operation ∞ and the closure operation #, assertion (2), and thus also (5), can be reformulated in several forms. Theorem 119 If in particular both R and S are nonvoid, then the following assertions are equivalent : #  (1) TR ⊆ FS ; (2) R ∧∞ ⊆ S ∧−1 ; (3)  X is topological openness reversing ; (4)  X is topological closedness reversing ; (5) for all A ⊆ X and R ∈ R ∧ there exists S ∈ S ∧ such that S −1 [ A ] ⊆ R ∞ [ A ]. Remark 70 Note that assertions (2) and (5) can again be reformulated in several forms. However, the operation ∧ is not inversion compatible. Therefore, the plausible notation ∨ = ∧ − 1 can be used.

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61

2.16 Some Further Results on The Identity Function Notation 10 In this section, more specially, we shall assume that R is a relator on X. Moreover, we shall consider the identity function  X of X as a function of X (R) to itself. Now, by Theorem 116, we can at once state the following Theorem 120 The following assertions are equivalent : (1) ER ⊆ DR ;

(2)  X is fatness reversing.

Remark 71 Thus, the hyperconnectedness of Steen and Seebach [182, p. 29], studied also by Levine [118] and about fifteen further authors mentioned in the References, is a particular case of fatness reversingness. Moreover, from Theorem 120 we can also see that paratopological connectedness introduced and investigated by Pataki and Száz [159] coincides with the fatness reversingness of the identity function whenever card (X ) > 1 and R = ∅. Thus, analogously to the results of [159, Section 15], we can also establish the following theorems. Theorem 121 The following assertions are equivalent : (1)  X is fatness reversing ; (2) Ac ∈ / ER for all A ∈ ER ;

(3) A ∩ B = ∅ for all A, B ∈ ER .

Proof From Theorem 120, we know that assertion (1) is equivalent to the inclusion ER ⊆ DR . Moreover, by using Theorems 45 and 46, we can easily see that this inclusion is equivalent to assertions (2) and (3). For instance, if (2) does not hold, then there exists A ∈ ER such that Ac ∈ / DR . This shows that ER  D R . ER . Hence, by Theorem 45, it follows that A ∈ Therefore, ER ⊆ DR implies (2), and thus (1) also implies (2). Theorem 122 The following assertions are equivalent : (1)  X is fatness reversing ; (2) A ∈ DR or Ac ∈ DR for all A ⊆ X ; (3) A ∈ DR or B ∈ DR whenever X = A ∪ B. Hint For instance if (3) does not hold, then there exist A, B ⊆ X such that X = A ∪ B, but A ∈ / DR and B ∈ / DR . Hence, by using Theorem 45, we can infer that Ac ∈ ER and B c ∈ ER . Moreover, we can also note that Ac ∩ B c = ( A ∪ B )c = X c = ∅. Therefore, by Theorem 121, assertions (1) does not also holds. This shows that (1) implies (3).

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Theorem 123 The following assertions are equivalent : (1)  X is fatness reversing ; (2) R (x) ∈ DR for all x ∈ X and R ∈ R ; (3) R (x) ∩ S (y) = ∅ for all x, y ∈ X and R, S ∈ R.   Hint To check this recall that B = R(x) : x ∈ X, R ∈ R is a base of the stack ER . Remark 72 In our unfinished work [188], the relator R is called semi-directed if (3) holds. Moreover, R is called quasi-directed if R (x) ∩ S (y) ∈ ER for all x, y ∈ X and R, S ∈ R. Note that thus “quasi-directed” implies “semi-directed” if ∅ ∈ / ER . That is, R is a total relator on X . Theorem 124 If card (X ) > 1 and R = ∅, then the following assertions are equivalent :

 −1 ∞  2  (1)  X is fatness reversing ; = X . (2) R   R  Remark 73 In our paper [159], the relator R is called properly well  former chained if R ∞ = X 2 . Moreover, R is called properly connected if the relator   RR −1 = R ∪ R −1 : R ∈ R ∞  is properly well-chained. That is, R  R −1 = { X 2 }. Thus, in particular, R is also called paratopologically well-chained (connected) if the relator R  is properly well-chained (connected). It has been shown that if card (X ) > 1 and R = ∅, then R is paratopologically well-chained (connected) if and only if ER = {X } ( ER ⊆ DR ). Thus, by Theorem 120, assertions (1) and (2) in Theorem 124 are also equivalent whenever card (X ) > 1 and R = ∅. Remark 74 The proper well-chainedness of R, in a detailed form, means only n that, for any x, y ∈ X and R ∈ R there exists a family ( xi )i=0 in X such that x0 = x, xn = y and ( xi−1 , xi ) ∈ R for all i = 1, 2, . . . , n. Thus, our former definition of proper well-chainedness is a straightforward generalization of Cantor’s chain-connectedness which was considered unsuitable by topologists. ( See, for instance, Thron [228, p. 29] and Wilder [238, p. 721].) Interestingly enough, our basic papers on well-chained and connected relators [113, 114, 159] have been roughly rejected by some leading topologists acting in the editorial boards of various mathematical journals. Moreover, Császár [58], noticing that the concept of connected “belongs rather to the theory of generalized topological spaces instead of topology in the strict sense”, did not quote our papers.

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63

Now, by Theorem 117, we can also at once state the following. Theorem 125 The following assertions are equivalent : (1) DR ⊆ ER ;

(2)  X is denseness reversing.

Moreover, analogously to Theorems 121 and 122, we can also easily prove the following two theorems. Theorem 126 The following assertions are equivalent : (1)  X is denseness reversing ; (2) Ac ∈ / DR for all A ∈ DR ; (3) A ∩ B = ∅ for all A, B ∈ DR . Theorem 127 The following assertions are equivalent : (1)  X is denseness reversing ; (2) A ∈ ER or Ac ∈ ER for all A ⊆ X ; (3) A ∈ ER or B ∈ ER whenever X = A ∪ B. Remark 75 From Theorem 126, we can see that irresolvableness, studied also by several authors, is a particular case of denseness reversingness. Following Hewitt [93], a topological space is called resolvable if it is the union of two disjoint dense subsets. ( For some generalizations, see also [56] and [175].)

2.17 Two Illustrating Examples and a Constancy Theorem The fact that the identity function of the real line R is continuous, but not contra continuous was already noted by Dontchev [63, Example 2.3]. Now, somewhat more generally, we can state the following Example 2 Let X = Rn and d be the usual metric on X . Then,  R = Rd = Brd :

 r >0 ,

where Brd = {(x, y) ∈ X 2 : d (x, y) < r }, is a tolerance relator on X , with several useful additional properties, such that, by considering  X as a function of X (R) to itself, we can state that (1) (2) (3) (4) (5)

X X X X X

is both proximal and topological interior reversing ; is neither proximal nor topological closure reversing ; is neither fatness nor denseness reversing ; is both proximal openness and closedness reversing ; is neither topological openness and closedness reversing.

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It is clear that x ∈ Brd (x)

Brd (x)\{x} = ∅

and

for all x ∈ X and r > 0. Therefore, by Corollaries 18 and 19, assertions (1) and (2) are true. Moreover, we can easily see that B1d (0) ∈ ER \DR ,

Qn ∈ DR \ER

and

B1d (0) ∈ TR \FR .

Therefore, by Theorems 116, 117 and 119, assertions (3) and (5) are also true. Now, to complete the proof, we need only prove assertion (4). For this, note that if x, y ∈ X and r > 0, then by choosing n ∈ N such that n −1 d (x, y) < r , and −1 using thepoints  xi = x + i n ( y − x ) with i = 0, 1, . . . , n, we can see that d n (x, y) ∈ Br . Therefore, X2 ⊆

∞  

Brd

n

 ∞ = Brd ,

n=0

 ∞ = X 2 . Therefore, the corresponding particular case of assertion (5) and thus Brd in Theorem 118 trivially holds. Remark 76 The proper well-chainedness of R can also be proved by using that each Brd is an absorbing translation relation on Rn [210]. By the results of our former paper [210], it is clear that, instead of the Euclidean space Rn , we could have considered here a non-degenerated vector relator space. By using two Sierpi´nski topologies on a two-point set, Dontchev [63, Example 2.5] also noted that an identity function can also be contra continuous, but not continuous. Now, somewhat more generally, we can state the following. Example 3 Let X = {0, 1 }, and R and S be two relations on X such that R (0) = {0},

R (1) = X

and

S (0) = X, S (1) = {1}.

Then, R = {R} and S = {S} are properly simple preorder relators on X , with S = R −1 , such that, by considering  X as a function of X (R) to X (S ), we can state that : (1) (2) (3) (4) (5)

X X X X X

is both proximal and topological interior reversing ; is neither proximal nor topological closure reversing ; is neither fatness nor denseness reversing ; is both proximal and topological openness reversing ; is both proximal and topological closedness reversing.

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65

By using the corresponding definitions, we can see that IntR (∅)  =  {∅} ;  IntR {0} = ∅, {0} ; IntR {1} = {∅} ; IntR (X ) = P(X ) ; ClR (∅)  =  ∅; ClR {0} = P(X )\{∅}  ; ClR {1} = {1}, X ; ClR (X ) = P(X )\{∅} ; intR (∅)  =  ∅; intR {0} = {0} ; intR {1} = ∅ ; intR (X ) = X ; clR (∅)  =  ∅; clR {0} = X ; clR {1} = {1} ; clR (X ) = X ;  ER = {0}, X ; DR = {0}, X ; τR = TR = P(X )\{1} ; τ-R = FR = P(X )\{0} ;

ClS (∅)  =  ∅ ;  ClS {0} = {0}, X ; ClS {1} = P(X )\{∅} ; ClS (X ) = P(X )\{∅} ; IntS (∅)  =  {∅} ; IntS {0} = {∅} ;  IntS {1} = ∅, {1} ; IntS (X ) = P(X ) ; clS (∅)  =  ∅; clS {0} = {0} ; clS {1} = X ; clS (X ) = X ; intS (∅)  =  ∅; intS {0} = ∅ ; intS {1} = {1} ; intS (X) = X ;  DS =  {1}, X ; ES = {1}, X ; τ-S = FS = P(X )\{1} ; τS = TS = P(X )\{0}.

Thus, the required assertions can, in principle, be easily verified by using the corresponding results of Sect. 2.15. Remark 77 This example can certainly be generalized by using two Pervin relators RA and RB with appropriate A , B ⊆ P(X ). Moreover, one may also be naturally investigate the contra continuity properties of the diversity relation cX and a Pervin relation R A with A ⊆ X . However, it is now more important to note that, as a straightforward extension of [64, Theorem 5.1] of Dontchev and Noiri, we can also prove the following. Theorem 128 If f is a function of a topologically connected relator space X (R) to a T1 -separated one Y (S ) such that its inverse relation f −1 is topological openness reversing, then f is a constant function. Proof Assume on the contrary that f is not constant. Then, there exist a, b ∈ X such that f (a) = f (b). Moreover, since Y (S ) is T1 -separated, for each y ∈ Y and z ∈ {y}c there exists S ∈ S such that y ∈ / S (z), and thus S (z) ⊆ {y}c . Hence, by Theorems 55 and 56, c we can see that {y} ∈ TS , and thus {y} ∈ FS . Therefore, by Theorem 110 and Definition 11, we can also state that f −1 [{y}] ∈ TR .

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Hence, it is clear that in particular we have   A = f −1 { f (a)} ∈ TR . Moreover, by using Remark 15 and Theorems 61 and 56, we can also see that  c   Ac = f −1 { f (a)} = f −1 { f (a)}c =

 y∈{

f −1 [ {y} ] ∈ TR ,

f (a)}c

and thus A ∈ FR . Therefore, A ∈ TR ∩ FR also holds. However, by [159, Theorem 15.9], we now have TR ∩ FR = { ∅, X }. Therefore, either A = ∅ or A = X hold. But, this is a contradiction since a ∈ A and b∈ / A. Remark 78 The following extensive References contain several, but far not all papers on contra continuity and hyperconnectedness. Papers which do not contain these terms in their titles, and those which are dealing with these subject in some unusual spaces, will be omitted. Moreover, several important papers on generalized uniform spaces will not also be listed here. These can mostly be found in the References of our former papers. Acknowledgements The author is greatly indebted to the anonymous referee for his careful pointing out more than one hundred errors in spelling, punctuation and numbering.

References 1. Acharjee, S., Rassias, Th.M., Száz, Á.: Upper and lower semicontinuous relations in relator spaces. In: Pardalos, P.M., Rassias, Th.M. (eds.) Analysis, Geometry, Nonlinear Optimization and Applications, Series in Computers and Operations Research, vol. 9, pp. 41–112. World Scientific (2023) 2. Ahmad, B., Noiri, T.: The inverse images of hyperconnected sets. Mat. Vesn. 37, 177–181 (1985) 3. Ajmal, N., Kohli, J.K.: Properties of hyperconnected spaces, their mappings into Hausdorff spaces and embeddings into hyperconnected spaces. Acta Math. Hung. 60, 41–49 (1992) 4. Akdag, M., Özkan, A.: Some properties of contra gb-continuous functions. J. New Res. Sci. 1, 40–49 (1992) 5. Aljarrah, H.H., Noorani, M.S.M., Noiri, T.: Contra ωβ-continuity. Bol. Soc. Paran. Mat. 32, 9–22 (2014) 6. Alli, K.: Contra g# p-continuous functions. Int. J. Math. Trends Technol. 4 (2013) 7. Al-Omari, A.A.: Contra continuity on weak structure spaces. Rend. Ist. Mat. Univ. Trieste 44, 423–437 (2012) 8. Al-Omari, A.A., Noiri, T.: A unified theory of contra-(μ, λ)-continuous functions in generalized topological spaces. Acta Math. Hungar. 135, 31–41 (2012) 9. Al-Omari, A.A., Noiri, T.: A unified theory of weakly contra-(μ, λ)-continuous functions in generalized topological spaces. Stud. Univ. Babes-Bolyai Math. 58, 107–117 (2013) 10. Al-Omari, A.A., Noorani, M.S.M.: Contra-ω-continuous and almost contra-ω-continuous. Int. J. Math. Math. Sci. Art. ID 40469, 13 pp (2007)

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194. Száz, Á.: Neighbourhood relators. Bolyai Soc. Math. Stud. 4, 449–465 (1995) 195. Száz, Á.: Relations refining and dividing each other. Pure Math. Appl. Ser. B 6, 385–394 (1995) 196. Száz, Á.: Topological characterizations of relational properties. Grazer Math. Ber. 327, 37–52 (1996) 197. Száz, Á.: Connectednesses of refined relators, Technical report. Institute of Mathematics, University of Debrecen, 6 pp (1996/14) 198. Száz, Á.: Uniformly, proximally and topologically compact relators. Math. Pannon. 8, 103–116 (1997) 199. Száz, Á.: An extension of Kelley’s closed relation theorem to relator spaces. Filomat (Nis) 14, 49–71 (2000) 200. Száz, Á.: Somewhat continuity in a unified framework for continuities of relations. Tatra Mt. Math. Publ. 24, 41–56 (2002) 201. Száz, Á.: Upper and lower bounds in relator spaces. Serdica Math. J. 29, 239–270 (2003) 202. Száz, Á.: An extension of Baire’s category theorem to relator spaces. Math. Morav. 7, 73–89 (2003) 203. Száz, Á.: Rare and meager sets in relator spaces. Tatra Mt. Math. Publ. 28, 75–95 (2004) 204. Száz, Á.: Galois-type connections on power sets and their applications to relators, Technical report. Institute of Mathematics, University of Debrecen, 38 pp. (2005/2) 205. Száz, Á.: Supremum properties of Galois-type connections. Comment. Math. Univ. Carolin. 47, 569–583 (2006) 206. Száz, Á.: Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities. Filomat (Nis) 21, 87–97 (2007) 207. Száz, Á.: Applications of fat and dense sets in the theory of additive functions, Technical report. Institute of Mathematics, University of Debrecen, 29 pp. (2007/3) 208. Száz, Á.: Galois type connections and closure operations on preordered sets. Acta Math. Univ. Comen. 78, 1–21 (2009) 209. Száz, Á.: Applications of relations and relators in the extensions of stability theorems for homogeneous and additive functions. Aust. J. Math. Anal. Appl. 6, 1–66 (2009) 210. Száz, Á.: Foundations of the theory of vector relators. Adv. Stud. Contemp. Math. 20, 139–195 (2010) 211. Száz, Á.: Galois-type connections and continuities of pairs of relations. J. Int. Math. Virt. Inst. 2, 39–66 (2012) 212. Száz, Á.: An extension of an additive selection theorem of Z. Gajda and R. Ger to vector relator spaces. Sci. Ser. A Math. Sci. (N.S.) 24, 33–54 (2013) 213. Száz, Á.: Lower semicontinuity properties of relations in relator spaces. Adv. Stud. Contemp. Math. (Kyungshang) 23, 107–158 (2013) 214. Száz, Á.: Inclusions for compositions and box products of relations. J. Int. Math. Virt. Inst. 3, 97–125 (2013) 215. Száz, Á.: A particular Galois connection between relations and set functions. Acta Univ. Sapientiae Math. 6, 73–91 (2014) 216. Száz, Á.: Generalizations of Galois and Pataki connections to relator spaces. J. Int. Math. Virtual Inst. 4, 43–75 (2014) 217. Száz, Á.: Basic tools, increasing functions, and closure operations in generalized ordered sets. In: Pardalos, P.M., Rassias, T.M. (eds.) Contributions in Mathematics and Engineering: In Honor of Constantin Caratheodory, pp. 551–616. Springer (2016) 218. Száz, Á.: Four general continuity properties, for pairs of functions, relations and relators, whose particular cases could be investigated by hundreds of mathematicians, Technical report, Institute of Mathematics, University of Debrecen, 17 pp. (2017/1) 219. Száz, Á.: Relationships between inclusions for relations and inequalities for corelations. Math. Pannon. 26, 15–31 (2017/18) 220. Száz, Á.: The closure-interior Galois connection and its applications to relational inclusions and equations. J. Int. Math. Virt. Inst. 8, 181–224 (2018)

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Chapter 3

The Continuous Representation Property in Utility Theory Juan C. Candeal

Abstract A topological space (X, τ ) satisfies the Continuous Representation Property (CRP) if every continuous total preorder defined on X can be represented by a continuous order-preserving real-valued function. The relevance of this property is discussed in the context of economics and social sciences. Certain characterizations of CRP are presented in terms of other familiar topological properties. In addition, two extensions of CRP, one regarding the semicontinuous case and the other involving an algebraic environment, are also discussed. Keywords Ordered structures on topological spaces · Countability axioms · Continuous representation property · Countable chain condition Classifications 54F05 · 06A05 · 91B16 · 91B14

3.1 Introduction The purpose of this article is two-fold. Firstly, the most significant results related to the so-called Continuous Representation Property1 (shortly, CRP) are reviewed. Secondly, we will present some new results concerning CRP that are linked to the familiar topological property called the Countable Chain Condition (briefly, CCC).

1 Although there are other expressions in the literature to refer to this property, such as those of a use-

ful topology, the continuous representability property,...etc.; following the suggestion of Professor G. B. Mehta, we have decided to call it the Continuous Representation Property as in [13]. J. C. Candeal (B) Departamento de Análisis Económico, Facultad de Economía y Empresa, Universidad de Zaragoza, Gran Vía 2. 50005, Zaragoza, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_3

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Because the article is mainly descriptive, a few proofs are only included leaving the interested reader the task of reading them in the references cited throughout the paper. CRP plays an important role within the topological approach to mathematical utility theory. The expression mathematical utility theory, that was used by Herden in [21], refers to the most significant mathematical results involving real-valued order-preserving2 functions that are relevant in economics and social sciences. There are different motivations to study CRP. From a topological point of view, CRP can be used to characterize other interesting properties such as order-extension properties of topological spaces or certain topological properties of Banach spaces in Functional Analysis (for details, see Yi [30], Campión et al. [7, 10], and Campión et al. [6]). From a more applied point of view, the study of the existence of utility functions turns out to be a very important issue in optimization theory and decision sciences. In economics, the beginning of the development and use of the theory of orderpreserving real-valued functions goes back to Debreu ([15]). Debreu’s work focuses on the existence of numerical representations of totally preordered sets endowed with a topology compatible with the order. In other words, it combines order and topology. In the context of measurement theory, there is also a vast literature related to the existence of order-preserving functions which satisfy some additional algebraic properties (see, for instance, [25]). On the one hand, CRP is closely related to what is called in the literature the continuous representation problem. Actually, CRP can be considered a step forward when studying the existence of continuous utility functions defined on a topological space (X, τ ) endowed with a total preorder . The resolution of the continuous representation problem has become a cornerstone in mathematical utility theory. A first and fundamental result in this direction was given by Debreu [16], who made use of his well-known open-gap lemma. Basically, Debreu’s result tells that a total preorder  defined on a topological space (X, τ ) admits a continuous order-preserving realvalued function if and only if it is continuous and representable. Continuity turns out to be a minimal (necessary) condition for the continuous representation problem to be met. On the other hand, it is widely recognized in the specialized literature that the two most important results of the topological approach to mathematical utility theory are Eilenberg’s theorem and Debreu’s theorem. Eilenberg’s theorem states that every continuous total preorder  defined on a connected and separable topological space (X, τ ) can be represented by a continuous utility function. Debreu’s theorem ensures the same conclusion as Eilenberg’s result, for every continuous total preorder, whenever connected plus separability are replaced with second countability. The abstraction of this property gave rise to the concept of a useful topology due to Herden [21]. Indeed, it should be noted that both Eilenberg’s theorem, as well as Debreu’s theorem, provide sufficient topological conditions on (X, τ ) so as to guarantee the existence of a continuous utility function for any continuous total preorder 2

In economics and decision sciences, it is very often to refer to a real-valued order-preserving function as a utility function. We will use indistinctly both terminologies.

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 defined on it. Herden [21] himself realized this fact and defined a topology τ on a set X to be useful whenever any continuous total preorder defined on X can be represented by a continuous utility function. Thus, for a useful topology, the continuous representation problem fulfils for all continuous total preorders. In this way, both Eilenberg’s theorem and Debreu’s theorem can be rephrased by saying that connected and separable, and, respectively, second countable topological spaces satisfy CRP. A short account of the contents of the article is now provided. After a preliminary section which includes certain basic definitions, we review in Sect. 3.3 the most significant results shown in the literature that characterize CRP. In particular, the seminal contributions of Herden [20, 21], Herden and Pallack [22], and Bosi and Herden [3] are presented. In addition, the simpler approach offered in Campión et al. [9, 10] to characterize CRP is also discussed. Both Herden and Pallack [22] and Bosi and Herden [3] show characterizations of CRP in terms of the fulfilment of the second countability axiom for certain topologies generated by complete separable systems. Complete separable systems can be considered as specializations of Nachbin-Urysohn-type families of open subsets which are related to separation axioms in ordered topological spaces (see also Burgess and Fitzpatrick [5]). In contrast, Campión et al. [9] provide a characterization of CRP in terms of the fulfilment of the second countability axiom for every preorderable subtopology of the given topology. This is a more direct way to deal with the problem that allows us to easily identify the familiar topological spaces that satisfy CRP. In Sect. 3.4, we pay special attention to the fulfilment of CRP regarding CCC. The case of metric spaces is discussed by presenting the important contribution of Estévez and Hervés [19]. Then, the most novel result of the paper is shown; to wit, in completely regular topological spaces CRP entails CCC. As a consequence, the following characterization result is reached: In separably connected and completely regular topological spaces, CRP amounts to CCC. Section 3.5 is devoted to introduce and discuss the two extensions of CRP. Firstly, we consider the corresponding property whenever continuity is replaced by semicontinuity in the definition of CRP. This leads to the so-called Semicontinuous Representation Property (shortly, SRP). Here, we present some important contributions that appear in Bosi and Herden [2]. In particular, the fact that SRP entails CRP is shown. Secondly, CRP, in the algebraic context of groups, is presented which leads to the Continuous Algebraic Representation Property (briefly, CARP). In this algebraic setting, the kind of utility representation we search for is also a group homomorphism. We provide a complete characterization of CARP and show that connected groups satisfy CARP. Section 3.6, which includes certain concluding remarks, ends the paper.

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3.2 Preliminaries A preorder  on an arbitrary nonempty set X is a reflexive and transitive binary relation defined on X . An antisymmetric preorder is said to be an order. A total preorder  on a set X is a preorder such that any pair of elements are comparable, i.e., for every x, y ∈ X either x  y or y  x (or both). If  is a total preorder on X , then the pair (X, ) is said to be a totally preordered set. If  is a total preorder on X , then the asymmetric relation ≺ associated with  is defined by [x ≺ y ⇐⇒ (x  y) ∧ ¬(y  x)]. The indifference relation ∼ associated with  is given by [x ∼ y ⇐⇒ (x  y) ∧ (y  x)]. It is easy to see that ∼ is an equivalence relation on X . Let (X, ) be a totally preordered set, and let X/ ∼ be the set of equivalence classes (i.e., the quotient space under ∼). If x ∈ X , then the notation [x] stands for the equivalence class of x. The total preorder  on X induces a total order  on X/ ∼ defined by [x]  [y] ⇐⇒ x  y. Let there be given, [x] and [y], two equivalence classes in X/ ∼. Then we say that the ordered pair ([x], [y]) is a jump if there is no [z] ∈ X/ ∼ such that [x] < [z] < [y], where < denotes the asymmetric part of . If ([x], [y]) is a jump, then we sometimes abuse notation and say that (x, y) is a jump in X . A totally preordered set (X, ) is said to be densely ordered if it has no jumps. A subset Z of X is said to be order-dense in X with respect to  if, for every x, y ∈ X with x ≺ y, there exists z ∈ Z such that x  z  y. (X, ) is said to be order-separable if it has a countable order-dense subset. If (X, ) is a totally preordered set, then a real-valued function u : X → R is said to be an order-preserving function if for every x, y ∈ X , [x  y ⇒ u(x) ≤ u(y)] and [x ≺ y ⇒ u(x) < u(y)]. If such a function u does exist, then  is said to be representable. We also refer to this function as a utility function or a numerical representation of . An order-isomorphism is a surjective order-preserving function. Let (X, τ ) be a topological space. A total preorder  on X admits (or can be represented by) a continuous utility function if there exists an order-preserving function for  that is continuous with respect to the topology τ and the usual (Euclidean) topology on the real line R. Let (X, ) be a totally preordered set. The family of all sets of the form L(x) = {a ∈ X ; a ≺ x} and G(x) = {a ∈ X ; x ≺ a}, where x runs over X , is a subbasis for a topology τ on X called the order topology. Let (X, τ ) be a topological space and let  be a total preorder on X . Then  is said to be continuous if, for each x ∈ X , the sets L(x) and G(x) are open in X . A topology τ on X is said to be preorderable if τ coincides with τ for some total preorder  defined on X . All definitions and concepts stated above can be seen in Bridges and Mehta [4] Definition 1 ([4, 17]) A topological space (X, τ ), or the topology τ , satisfies: (i) CRP (the Continuous Representation Property) if every continuous total preorder defined on X can be represented by a continuous utility function.

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(ii) CCC (the Countable Chain Condition) if every family of pairwise disjoint nonempty open subsets of X is countable.

3.3 The Continuous Representation Property: A Review of the Main Contributions In this section, the main contributions regarding different approaches to CRP that appeared in the recent literature are revised. In particular, those of Herden [21], Herden and Pallack [22], Campión et al. [9, 10], and Bosi and Herden [3]. As already mentioned in Sect. 3.1, CRP, under the name of a useful topology, appears for the first time in Herden [21] as an abstract way of viewing both Eilenberg’s theorem and Debreu’s theorem. The approach followed by Herden in [21], and continued in Herden and Pallack [22], is based on the concept of an R-separable system. Let us recall this important concept in the next definition. As usual, if (X, τ ) is a topological space, then A denotes the topological closure of the set A ⊆ X . Definition 2 ([20, 22]) Let (X, τ ) be a topological space endowed with a binary relation R. A family E of open R-decreasing subsets of X is said to be an R-separable system on X if it satisfies the following conditions: (i) There are sets E 1 , E 2 ∈ E such that E 1 ⊂ E 2 . (ii) For all sets E 1 , E 2 ∈ E such that E 1 ⊂ E 2 there exists E 3 ∈ E so that E 1 ⊂ E 3 ⊂ E3 ⊂ E2 . Remark 1 (i) A subset A ⊆ X is R-decreasing if x ∈ A and yRx, entails y ∈ A, for every x, y ∈ X . (ii) If R is the equality relation “=” on X , i.e., the discrete order on X , then an R-separable system on X is simply called a separable system on X . (iii) The concept of an R-separable system is closely related to that of a decreasing scale as studied by Burgess and Fitzpatrick in [5]. (iv) Let E be an R-separable system on X . If for any pair of distinct subsets E 1 , E 2 ∈ E, at least one of the inclusions E 1 ⊂ E 2 or E 2 ⊂ E 1 holds, then E is said to be a linear R-separable system.3 It is not difficult to see that every R-separable system E on X contains some linear R-separable system. Moreover, every linear R-separable system E induces a total preorder  on X that extends R and such that the order topology of  is coarser than the topology generated by E. In particular,  is continuous (for details, see Herden [21] and Herden and Pallack [22]). Linear, or complete, separable systems on X are closely related to continuous total preorders defined on X . In order to see that, let us denote by P() the class of all continuous total preorders defined on X and by SC (X ) the class of all linear separable systems on X . For each linear separable system E, consider the topology 3

A linear separable system on X is called in Bosi and Herden [3] a complete separable system.

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generated by the elements E and X \ E, E ∈ E. Denote this topology by tE . On SC (X ), the following binary relation ∼ can be defined: E ∼ L ⇔ tE = tL . It is not difficult to prove that ∼ so-defined is an equivalence relation on SC (X ). Denote by SC (X )|∼S the corresponding quotient space under ∼. Then, the following interesting result is shown in Bosi and Herden [3]. Theorem 1 ([3]) Let (X, τ ) be a topological space. Then there is a one-to-one correspondence between P() and SC (X )|∼S . As far as we know the first characterization of CRP was given in Herden and Pallack [22]. This characterization was established in terms of a countable-chaintype condition, for particular families of open and closed subsets, together with certain countability properties of linear separable systems. In order to state it, some definitions and concepts are still needed. Definition 3 ([22]) A topological space (X, τ ) is said to satisfy the Open-closed Countable Chain Condition (briefly, OCCC) if every family F of open and closed subsets that satisfies the following two conditions is countable:  (OC1): Either F ⊂ F  or F  ⊂ F, for all F,  F ∈ F. F  F  (OC2): For every F ∈ F it holds that F  F,F  ∈F

 F  F  ,F  ∈F

Remark 2 Note that OCCC is a natural condition (actually, it is a necessary condition) for CRP to be met because it eschews the existence of an uncountable number of jumps for any continuous total preorder defined on X . Indeed, let  be a total preorder on X and let [ai , bi ]i∈I denote the set of jumps of . Assume that τ is the order topology induced by . Then, it is simple to see that the family F = (L(bi ))i∈I is a family of open and closed subsets of X that clearly satisfies conditions (OC1) and (OC2). Thus, if  admits a utility function, then, necessarily, F has to be countable. Therefore, OCCC is a necessary condition for CRP to be held in this context. In the general case, it is proved in Herden and Pallack [22] that if (X, τ ) does not satisfy OCCC, then it is possible to construct a continuous total preorder on X that has uncountably many jumps, hence it is non-representable, so violating CRP. Let now E be a linear separable system and consider all the pair (B, E) ∈ E × E for which there is C ∈ E such that B  C ⊂ C  E. Denote by Z(E) the set of all these pairs. Definition 4 ([22]) Let (X, τ ) be a topological space. A linear separable system E of X is said: (i) to have a countable refinement if there is a countable family O of open sets of X such that for every pair (B, E) ∈ Z(E) there is O ∈ O such that O ⊂ E ∩ X \ B, (ii) to be second countable if there is a countable subset H of E such that for every pair (B, E) ∈ Z(E) there is H + ∈ H such that B ⊂ H + ⊂ H + ⊂ E.

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 In addition, denote by G(E) the set of all (open) sets E ∈ E for which BE,B∈E B  E. Let GG stand for the subset of SC (X ) which consists of all linear separable systems on X , such that G(E) is countable. Then, the following characterizations of CRP hold. Theorem 2 ([22]) Let (X, τ ) be a topological space. Then the following assertions are equivalent: (i) τ satisfies CRP, (ii) τ satisfies OCCC and every linear separable system E ∈ SC (X ) on X has a countable refinement, (iii) τ satisfies OCCC and every linear separable system E ∈ SC (X ) on X is second countable, (iv) τ satisfies OCCC and every linearly ordered subtopology of τ that is induced by some linear separable system E ∈ GG on X is second countable. Obviously, if (X, τ ) is connected, then OCCC can be omitted and the characterizations of CRP simplify somewhat. Corollary 1 ([22]) Let (X, τ ) be a connected topological space. Then the following assertions are equivalent: (i) (ii) (iii) (iv)

τ satisfies CRP, Every linear separable system E ∈ SC (X ) on X has a countable refinement, Every linear separable system E ∈ SC (X ) on X is second countable, Every linearly ordered subtopology of τ that is induced by some linear separable system E ∈ GG on X is second countable.

Theorem 2 provides distinct interesting theoretical characterizations of CRP. However, they are not easy to apply in practise because they demand the control of all linear separable systems defined on (X, τ ). The approach pursued in Campión et al. [9] is slightly different and is based upon the following ideas. Firstly, it should be noted that, in order to obtain a characterization of CRP, we only need to have control of all subtopologies of τ that are induced by total preorders defined on X . Each one of such a topology will be called a preorderable subtopology on X . Now, because the fullfilment of CRP requires the existence of a utility function for each one of such total preorders, a characterization of the existence of such a function will be needed. Fortunately, this has been extensively studied in the literature and such a characterization amounts to the fullfilment of the second countability axiom of the order topology induced by the corresponding total preorder (see, e.g., Bridges and Mehta [4]). Secondly, and as a consequence of the previous argument, we are interested in studying which topological properties on (X, τ ) make these order topologies, induced by the corresponding total preorders, to be second countable so that their continuous representation can be guaranteed. These topological properties on (X, τ ) will be mainly based on covering axioms and countability axioms. In addition, note that all of these axiomatics have been studied for a long time in the context of totally ordered topological spaces (see, e.g., Lutzer and Bennet [26]).

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We now collect these remarks in the following result that appears in Campión et al. [9]. Theorem 3 ([9]) A topological space (X, τ ) satisfies CRP if and only if all its preorderable subtopologies are second countable. Remark 3 Another characterization of CRP can be easily given in terms of the fullfilment of both separability and pseudometrizability for each of its preorderable subtopologies (for details, see [9]). The following lemma can be seen in Campión et al. [9] and Herden and Pallack [22]. Recall that a topological space (X, τ ) is said to be separably connected if for every two points a, b ∈ X there exists a connected and separable subset Ca,b ⊆ X that includes a and b. In addition, (X, τ ) satisfies the Lindelöf property if every open covering of X has a countable subcovering. Lemma 1 ([9, 22]) Let (X, τ ) be a topological space. (i) Assume τ = τ , for some total preorder  on X . If (X, τ ) is connected and separable, then it is second countable. Moreover, second countability of (X, τ ) is implied in any of the following cases: (a) If it is separably connected and satisfies Lindelöf’s property, (b) If it is separably connected and satisfies CCC, (ii) If (X, τ ) is second countable, then so (X, τ ) is, for every continuous total preorder  on X . This simple approach allows us for recovering (and generalizing) the two most important results of mathematical utility theory; to wit, Eilenber’g theorem and Debreu’s theorem as well as to obtain new results that entail the fullfilment of CRP for familiar classes of topological spaces. From an applied point of view this is important because, for example, Banach spaces and, more generally, convex spaces are frequently studied in economics and decision sciences. In the infinite-dimensional case, these spaces may fail to be second countable or separable. This means that the continuous representation problem in these spaces is not guaranteed by either Eilenberg’s theorem or Debreu’s theorem. Thus, distinct characterizations of CRP are of particular interest in mathematical utility theory. Corollary 2 ([10]) A topological space (X, τ ) satisfies CRP provided that: (i) (ii) (iii) (iv)

it is connected and separable, it is second countable, it is separably connected and satisfies Lindelöf’s property, it is separably connected and satisfies CCC.

Proof Statements (i) to (iv) are straightforward consequences of Theorem 3 and Lemma 1. Moreover, note that connectedness, separability, Lindelöf’s property, and CCC are topological properties that are inherited by subtopologies. 

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Remark 4 It should be noted that none of the statements, (i) to (iv), that appears in Corollary 2, can be reversed. Counterexamples to these statements are {0, 1} × ∞ , where {0, 1} is endowed with the discrete topology and ∞ is equipped with the weak topology, for the cases (i) and (iii). A counterexample to (ii) is given by the Niemitzki plane, and a counterexample to (iv) is the product space {0, 1} × X , where {0, 1} is endowed with the discrete topology and X stands for the Alexandroff topology on [0, 1] × [0, 1]. Note that [0, 1] × [0, 1], endowed with the Alexandroff topology, is a compact and path-connected topological space (hence, by Corollary 2(iii) above, it satisfies CRP). However, it does not satisfy CCC and fails to be first countable (see Steen and Seebach [29] and Campión et al. [9] for details).

3.4 CRP and CCC The results presented in the previous Sect. 3.3 suggest that CRP is closely related to the fullfilment of some kind of countable chain condition. However, and as already mentioned there, the conditions shown in Theorem 2 and other related results are not easy to check in practise. In this section, we intend to make a further contribution by showing certain characterizations of CRP involving familiar topological properties of the given topological space (X, τ ). In particular, we will focus on the role played by CCC in connection with CRP. In addition, and in certain particular settings, we will be able to explicitly construct continuous total preorders which will allow us for proving that CCC is a necessary condition for CRP to be satisfied. Our approach will be based upon an important result due to Estevéz and Hervés [19] in the context of metric spaces. Let us recall Estevéz and Hervés’ theorem. Theorem 4 ([19]) Let (X, τ ) be a nonseparable metric space. Then there is a continuous total preorder on X that does not admit a utility function. The proof provided in Estévez and Hervés’ paper is based upon the following nice idea. Under the conditions of the theorem, it is possible to construct a continuous function from X into the long line L, equipped with its order topology, whose range fills ω1 .4 Then, the binary relation defined on X is the one given by the corresponding pre-images of the elements in L. Clearly, the binary relation so-defined turns out to be a continuous total preorder on X that does not admit a utility function. As noted in Candeal et al. [11], a straightforward consequence of Estévez and Hervés’ theorem is the following one. Corollary 3 ([11]) A metric space satisfies CRP if and only if it is separable.

As usual ω1 stands for the first uncountable ordinal number. The long line L is the lexicographic product of ω1 and [0, 1), both endowed with their natural orders. Alternatively, L can be defined in the following way: Between each ordinal α and its successor α + 1 put one copy of the real interval (0, 1). Then L is the space obtained in this way endowed with its natural order.

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Remark 5 Because in metric spaces separability, second countability and CCC are equivalent conditions, Corollary 3 can be rephrased in the following terms: A metric space satisfies CRP if and only if it satisfies CCC. Corollary 3 covers the cases of normed linear spaces and Banach spaces that so often appear in practise. Some other examples where Corollary 3 applies are collected in the next result. Corollary 4 ([10]) Let (G, τ ) be a topological group and denote by (C0 (G), ω) the space of continuous complex functions defined on G which vanish at infinity equipped with the weak topology ω. In the following cases, CRP and CCC are equivalent conditions for (G, τ ): (i) If (G, τ ) is first countable. (ii) If (G, τ ) is locally compact and (C0 (G), ω) satisfies the Lindelöf property. (iii) If (G, τ ) is locally compact and (C0 (G), ω) is normal. Proof (i) is a direct consequence of Corollary 3 and the classical Birkhoff-Kakutani theorem (see Birkhoff [1] or Kakutani [24]) which states that a topological group (G, τ ) is first countable if and only if it is metrizable. In a similar manner, (ii) and (iii) are consequences of Corollary 3 in combination with the following result that appears in Corson [14]: A locally compact topological group (G, τ ) is metrizable if and only if (C0 (G), ω) satisfies the Lindelöf property if and only if (C0 (G), ω) is a normal space.  Remark 6 It should be noted that CCC can be replaced by separability, or second countability, in the assertions of Corollary 4 because (G, τ ) turns out to be metrizable. However, this is not the case for a general topological group. To cite an example, consider the Hilbert space 2 (R) endowed with the weak topology ω. It is well-known that (2 (R), ω) is, in particular, a nonseparable topological group. Nevertheless, and as proved in Campión et al. [7], (2 (R), ω) satisfies CRP. We now recall the definition of a completely regular topological space. Definition 5 ([17]) A topological space (X, τ ) is said to be completely regular provided that for every closed set C ⊂ X and every point x ∈ / C there is a continuous function f : X → [0, 1] such that f (x) = 0 and f |C = 1. Remark 7 The following equivalent statement of a completely regular topological space will be helpful later on: A topological space (X, τ ) is completely regular provided that for every open set U ⊂ X , there is an open set V ⊆ U, a point x ∈ V, and a continuous function g : X → [0, 1], such that g(x) = 1 and g|V c = 0. Note that this assertion follows directly from Definition 5 by considering and open set V ⊆ U that includes x and the continuous function g : X → [0, 1] defined as g(t) = 1 − f (t), t ∈ X , where f : X → [0, 1] is continuous and such that f (x) = 0 and f |V c = 1. Now we are able to present the main novel contribution of the current article. It should be noted that Theorem 5 below generalizes Corollary 6.2 of Herden and Pallack [22] where a similar result is stated but only for locally finite families of pairwise disjoint open subsets of X .

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Theorem 5 Let (X, τ ) be a completely regular topological space that satisfies CRP. Then it satisfies CCC. Proof Suppose, by way of contradiction, that (X, τ ) does not satisfy CCC. Let us then show that there is a continuous total preorder defined on X which is not representable. This will imply that τ does not satisfy CRP. To that end, let (Uα )α∈ω1 be an uncountable family of pairwise disjoint nonempty open subsets of X . For each α ∈ ω1 , pick up a point xα ∈ Uα . Because (X, τ ) is completely regular, by Remark 7 for each α ∈ ω1 there is an open set containing xα , say Vα , and a continuous function f α : X → [0, 1] such that f α (xα ) = 1 and f α |(Vα )c = 0. Without loss of generality we can assume Vα = Uα . Now, note that, for each α ∈ ω1 , the long line interval [0, α] ⊂ L is orderisomorphic (hence homeomorphic) to the real interval [0, 1]. Denote this orderisomorphism by φα . In particular, it holds that φα (0) = 0 and φα (α) = 1. Define then the function u : X → L as follows:  −1 φα ( f α (x)) if x ∈Uα u(x) = 0 if x ∈ X \ α∈ω1 Uα It should be observed that u is well-defined and also that u(xα ) = α, for every α ∈ ω1 . Let us now prove that u is a continuous function where continuity refers to the topology τ on X and the order topology on L. To that end, let b ∈ L. Then, it is sufficient to see that u −1 (L(b)) and u −1 (G(b)) are open subsets of X . First, note that u −1 (L(0)) = u −1 (∅) = ∅ ∈ τ . For each α ∈ ω1 , denote by Bα = {x ∈ ∈ X : f α (x) > Uα : φ−1 α ( f α (x)) > b}. Then, Bα = {x ∈ Uα : f α (x) > φα (b)} = {x  φα (b)}, which is open because f α is continuous. Thus, u −1 (G(b)) = α∈ω1 {x ∈ X :   φ−1 α ( f α (x)) > b} = α∈ω1 {x ∈ X : f α (x) > φα (b)} = α∈ω1 Bα , which is open because is the union of open sets. Similarly, for each α ∈ ω1 , denote by Aα = {x ∈ ( f α (x)) < b}. Then, Aα = {x ∈ Uα : f α (x) < φα (b)} = {x ∈ X : f α (x) < Uα ; φ−1 α −1 (b)} Uα , which is open because f φ α is continuous and Uα ∈ τ . Thus, u (L(b)) = α −1 {x ∈ X : φ ( f (x)) < b} = A ∈ τ . α α α∈ω1 α∈ω1 α To finish the proof, consider the binary relation on X defined as follows: x u y if and only if u(x) ≤ u(y), (x, y ∈ X ). It is straightforward to see that u is a continuous total preorder on X which is non-representable because ω1 ⊆ u(X ). Therefore, (X, τ ) does not satisfy CRP and the desired contradiction is reached.  As a strengthening of Corollary 4, the following result holds. Corollary 5 Let (X, τ ) be a topological space. Then in any of the following cases, CRP entails CCC: (i) If (X, τ ) is locally compact and Hausdorff, (ii) If (X, τ ) is normal. Proof Both statements follow directly from Theorem 5 since it is well-known that locally compact Hausdorff topological spaces and normal spaces are both completely regular (see [17]). 

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For the particular case of separably connected and completely regular topological spaces, we have the following characterization of CRP. Theorem 6 For a separably connected and completely regular topological space, CRP amounts to CCC. Proof By Theorem 5, it suffices to show that CCC entails CRP. Suppose that (X, τ ) satisfies the CCC and let  be a continuous total preorder defined on X . Note that it is sufficient to prove that  admits a utility function. Since (X, τ ) is a separably connected topological space, as a result of Campión et al. [6], there is a continuous order-preserving function u : X → L, where L denotes the long line. Note that u(X ) is an interval of L. Furthermore, there is an ordinal α0 ∈ ω1 that bounds u(V ) (i.e., α ≤ L α0 , for all α ∈ u(X ), where ≤ L stands for the natural total order defined on L). Indeed, suppose, otherwise, that u(X ) exhausts L. Then by considering (Uα )α∈ω1 = (u −1 (α, α + 1))α∈ω1 , we would obtain an uncountable family of pairwise disjoint nonempty open subsets of X , which would contradict CCC. So, there is a countable ordinal that bounds u(X ), and therefore, u(X ) can be identified with a subset of the real line. This means that  is representable by a utility function, hence (X, τ ) satisfies CRP.  As a consequence of the previous theorem, we present the following corollary that includes certain results shown in Campión et al. [7] and Candeal et al. [12]. Corollary 6 ([7, 12]) Let (V, +, ·R , τ ) be a locally convex topological real vector space. Then τ satisfies CRP if and only if it satisfies CCC. In particular, the weak topology of a Banach space satisfies CRP.

3.5 Further Generalizations of CRP In this section, we briefly present some results that generalize CRP in certain environments. Firstly, we study the consequences of relaxing the continuity requirement, that appears in the definition of CRP, by replacing it with semicontinuity. This leads to the so-called Semicontinuous Representation Property (in short, SRP).5 Secondly, we will pay attention to the fulfilment of CRP in certain algebraic settings; in particular, in groups. In this situation, we will consider continuous utility functions that are also group homomorphisms which will lead us to the Continuous Algebraic Representation Property (in short, CARP). We begin with some basic concepts and definitions. Let X be a nonempty set endowed with a total preorder . The upper topology, u , is the one generated by all the sets of the form L(x) = {a ∈ X : a ≺ denoted by τ x} (x ∈ X ). 5

The semicontinuous representation property was introduced by Bosi and Herden in [2] under the name of a completely useful topology.

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Definition 6 ([2, 8]) Let (X, τ ) be a topological space. A total preorder  defined on X is said to be upper semicontinuous (respectively, lower semicontinuous) if L(x) ∈ τ , (respectively, G(x) ∈ τ ) for every x ∈ X . Remark 8 (i) If X is a set endowed with a total preorder , then the lower topology, l denoted by τ , is the one generated by all the sets of the form G(x) = {a ∈ X : x ≺ a} (x ∈ X ). (ii) Note that a total preorder  defined on X is upper semicontinuous if and only u if τ ⊆ τ . Similarly, a total preorder  defined on X is lower semicontinuous if and

l only if τ ⊆ τ. (iii) Throughout this section, we will focus on upper semicontinuous (for brevity, semicontinuous) total preorders on X . Nevertheless, all the results stated below extend straightforwardly to the lower semicontinuous case.

The semicontinuous representation problem, which parallels that of the continuous representation problem, concerns the existence of a semicontinuous utility function for a total preorder defined on a topological space. The most important result in this context is due to Rader [28], who proved that, in a second countable topological space, every semicontinuous total preorder admits a semicontinuous utility function. The fact of relaxing the continuity, and replacing it with semicontinuity, is important in practise because in many optimization problems the study of the existence of maximal elements, according to certain total preorders defined on a topological space, is required. To this respect, it should be noted that these maximals are guaranteed whenever the given total preorder is (upper) semicontinuous and the topological space satisfies certain topological conditions (for instance, compactness). As Eilenberg’s theorem and Debreu’s theorem were the motivations for introducing CRP, Rader’s theorem can be considered to be the natural inspiration for studying SRP. Let us present now this topological property. Definition 7 ([2, 8]) A topological space (X, τ ), or the topology τ , satisfies SRP provided that every semicontinuous total preorder on X can be represented by a semicontinuous utility function. Remark 9 Note that, according to Rader’s result, it holds that every second countable topological space satisfies SRP. Let X be a nonempty set. A topology τ on X is said to be upper preorderable if u it coincides with the upper topology of a total preorder defined on X (i.e., if τ = τ , for some total preorder  on X ). Upper preorderable subtopologies of τ play an important role in characterizing SRP similarly to the one played by preorderable subtopologies of τ when characterizing CRP (see Theorem 3 above and Theorem 4.2 in Campión et al. [8]). Theorem 7 ([8]) A topological space (X, τ ) satisfies SRP if and only if all its upper preorderable subtopologies are second countable.

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Theorem 7 has the following somewhat surprising consequence (see Bosi and Herden [2] and Campión et al. [8]). Corollary 7 ([2, 8]) SRP implies CRP. Remark 10 (i) The claim of Corollary 7 lies on the fact that, for a total preorder  u on a set X , it holds that τ is second countable if and only if so τ is (see Theorem 3.3 in Campión et al. [8]). (ii) The converse of Corollary 7 is not true in general. To see that consider the first uncountable ordinal ω1 equipped with the dual order ≤d (i.e., α ≤d β if and only if β ≤ α, for all α, β ∈ ω1 , where ≤ denotes the natural order on ω1 ). Let τ≤u d be the upper topology on ω1 . Then, the usual total order ≤ on ω1 is (upper) semicontinuous with respect to τ≤u d , hence τ≤u ⊆ τ≤u d . Moreover, it is not difficult to see that the only continuous, with respect to τ≤u d , total preorder  on ω1 is the trivial one (i.e., α ∼ β, for all α, β ∈ ω1 ). This means that, trivially, (ω1 , τ≤u d ) satisfies CRP. However, it does not satisfy SRP because, obviously, τ≤u d is an upper preorderable subtopology of τ≤u d which fails to be second countable. In the seminal paper by Bosi and Herden [2], different characterizations of SRP are provided which are reminiscent of Theorem 2 above in the semicontinuous setting. In order to illustrate such characterizations, some new concepts are still needed. Let (X, τ ) be a topological space and consider the set O which consists of families of all open subsets of X that are linearly ordered by set inclusion. A typical member of O"  O will be denoted by O. A set O ∈ O is said to be isolated provided that OO"O  O O  . A topology τ on X for which every O ∈ O only has countably many OO  ∈O

isolated sets is said to be countably isolated. In a similar way, the concepts of a left-isolated set and a right-isolated set are defined. Let O ∈ O be arbitrarily chosen. a set O ∈ O is saidto be right isolated  Then (respectively, left isolated) if O  O  (respectively, if O"  O). The OO  ∈O

OO"O

family O ∈ O is said to be semi-separable if there is a countable subfamily O of O such that for every right isolated set O ∈ O and every O" ∈ O such that O"  O there exists some set O  ∈ O in such a way that O"  O   O. Then the following result is reached (see Theorem 3.2 in Bosi and Herden [2]). Theorem 8 ([2]) For a topological space (X, τ ), the following assertions are equivalent: (i) (X, τ ) satisfies SRP, (ii) Every linearly ordered subtopology of τ is semi-separable, (iii) τ is countably isolated and every linearly ordered subtopology of τ that only contains countably many left isolated sets is second countable, (iv) Every linearly ordered subtopology of τ that has a base that only contains countably many sets that are not right isolated is second countable.

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As seen above, for a topological space CRP is, in general, weaker than SRP. However, the converse also holds in certain significant cases as the next result shows. Theorem 9 ([8, 10]) Let (X, τ ) be a topological space. Then SRP and CRP are equivalent properties topological in the following cases: (i) If (X, τ ) is metrizable. (ii) If (X, τ ) is a totally preordered set. Remark 11 (i) Actually, in the two cases above, SRP amounts to second countability of (X, τ ). (ii) For the weak topology ω of a Banach space (X, || · ||) we have the following result (see Theorem 5.1 in Campión et al. [8]): (X, ω) satisfies SRP if and only if (X, || · ||) satisfies CCC. We conclude this section by presenting certain results concerning CRP in an algebraic environment. In particular, we will concentrate on groups (see Candeal et al. [12] and Campión et al. [10]). First, a basic definition is needed. Definition 8 ([10, 12]) Let (G, +) be a group, with identity e ∈ G, and let  be a total preorder defined on G. Consider the order topology τ on G. (i)  is said to be translation-invariant provided that x  y =⇒ x + z  y + z and z + x  z + y, for every x, y, z ∈ G. (ii)  admits (or can be represented by) an additive utility function if there is an order-preserving function u : X → R which, in addition, satisfies u(x + y) = u(x) + u(y), for every x, y ∈ G.  (iii) A basis of the identity for τ , say Ve , is said to be absorbing if n∈N n B = G, for every B ∈ Ve . Now we introduce the algebraic version of the continuous representation property for groups. Definition 9 ([12]) Let (G, +) be a group equipped with a topology τ . Then (G, τ ), or the topology τ , satisfies the Continuous Algebraic Representation Property (shortly, CARP) if every continuous and translation-invariant total preorder  defined on G admits a continuous and additive utility function. Remark 12 Note that the semicontinuous version of CARP, which could be abbreviated by SARP, in this context turns out to be equivalent to CARP. Indeed, it can be easily proved that any upper semicontinuous and translation-invariant total preorder defined on G is, in fact, continuous. The following characterization of CARP is in order (see Candeal et al. [12]). Theorem 10 ([12]) Let (G, +) be a group endowed with a topology τ . Then the following assertions are equivalent:

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(i) (G, τ ) satisfies CARP, (ii) For every continuous and translation-invariant total preorder  on G, there is an absorbing basis of the identity for τ . An important case for which CARP holds is given in the following theorem which generalizes a result due to Iseki [23] (see also [27]). Theorem 11 ([12]) Let (G, +) be a group equipped with a connected topology τ . Then (G, τ ) satisfies CARP.

3.6 Concluding Remarks In this article, the so-called Continuous Representation Property (CRP) on topological spaces has been reviewed. This property is closely related to the continuous representation problem which turns out to be a core issue within the theory of orderpreserving real-valued functions. We present the most important results appeared in the literature in relation with CRP; in particular, those by Herden and Pallack [22] and Bosi and Herden [3]. The significance of these approaches in economics and social sciences has also been discussed. We have paid special attention to the Countable Chain Condition (CCC) by showing that, in completely regular topological spaces, CRP entails CCC. Moreover, in separably connected and completely regular topological spaces, CRP amounts to CCC. This result is the main novel contribution of the paper. In addition, we have also revised CRP in the semicontinuous case, which leads to the so-called Semicontinuous Representation Property (SRP), and considered the extension of CRP in certain algebraic environments.

References 1. Birkhoff, G.: A note on topological groups. Compos. Math. 3, 427–430 (1936) 2. Bosi, G., Herden, G.: On the structure of completely useful topologies. App. Gen. Topol. 3(2), 145–167 (2002) 3. Bosi, G., Herden, G.: The structure of useful topologies. J. Math. Econ. 82, 69–73 (2019) 4. Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. Springer, Berlin (1995) 5. Burgess, D.C.J., Fitzpatrick, M.: On separation axioms for certain types of ordered topological space. Math. Proc. Camb. Philos. Soc. 82, 59–65 (1977) 6. Campión, M.J., Candeal, J.C., Granero, A.S., Induráin, E.: Ordinal representability in Banach spaces. In: Castillo, J.M.F., Johnson, W.B. (eds.) Methods in Banach space theory, pp. 183–196. Cambridge University Press, UK (2006) 7. Campión, M.J., Candeal, J.C., Induráin, E.: The existence of utility functions for weakly continuous preferences on a Banach space. Math. Soc. Sci. 51, 227–237 (2006) 8. Campión, M.J., Candeal, J.C., Induráin, E.: Semicontinuous order-representability of topological spaces. Bol. Soc. Mat. Mex. 15(3), 81–89 (2009a)

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9. Campión, M.J., Candeal, J.C., Induráin, E.: Preorderable topologies and order-representability of topological spaces. Topol. Appl. 159, 2971–2978 (2009b) 10. Campión, M.J., Candeal, J.C., Induráin, E., Mehta, G.B.: Continuous order representability properties of topological spaces and algebraic structures. J. Korean Math. Soc. 49(3), 449–473 (2012) 11. Candeal, J.C., Hervés, C., Induráin, E.: Some results on representation and extension of preferences. J. Math. Econ. 29, 75–81 (1998) 12. Candeal, J.C., Induráin, E., Sanchis, M.: Order representability in groups and vector spaces. Expo. Math. 30, 103–123 (2012) 13. Candeal, J.C.: The existence and the non-existence of utility functions in order-theoretic, algebraic and topological environments. In: Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E. (eds.) Mathematical Topics on Representations of Ordered Structures and Utility Theory: Essays in Honor of Professor Ghanshyam B. Mehta, Studies in Systems, Decision and Control, pp. 23–45. Springer, Switzerland (2020) 14. Corson, H.H.: The weak topology of a Banach space. Trans. Am. Math. Soc. 101, 1–15 (1961) 15. Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R., Coombs, C., Davies, R. (eds.) Decision Processes, pp. 159–165. Wiley, New York (1954) 16. Debreu, G.: Continuity properties of Paretian utility. Int. Econ. Rev. 5, 285–293 (1964) 17. Dugundji, J.: Topology. Allyn and Bacon, Boston (1966) 18. Eilenberg, S.: Ordered topological spaces. Am. J. Math. 63, 39–45 (1941) 19. Estévez, M., Hervés, C.: On the existence of continuous preference orderings without utility representation. J. Math. Econ. 24, 305–309 (1995) 20. Herden, G.: On the existence of utility functions II. Math. Soc. Sci. 18, 109–117 (1989) 21. Herden, G.: Topological spaces for which every continuous total preorder can be represented by a continuous utility function. Math. Soc. Sci. 22, 123–136 (1991) 22. Herden, G., Pallack, A.: Useful topologies and separable systems. App. Gen. Topol. 1(1), 61–82 (2000) 23. Iseki, K.: On simple ordered groups. Port. Math. 10(2), 85–88 (1951) 24. Kakutani, S.: Über die Metrisation der topologischen Gruppen. Proc. Imp. Acad. (Tokyo, 1912) 12, 82–84 (1936) 25. Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement: Additive and Polynomial Representations, vol. I. Academic Press, New York (1971) 26. Lutzer, D.J., Bennet, H.R.: Separability, the countable chain condition and the Lindelöf property on linearly ordered spaces. Proc. Am. Math. Soc. 23, 664–667 (1969) 27. Montgomery, D.: Connected one dimensional groups. Ann. Math. 40(1), 195–204 (1948) 28. Rader, T.: The existence of a utility function to represent preferences. Rev. Econ. Stud. 30(1), 229–232 (1963) 29. Steen, L.A., Seebach, J.A.: Counterexamples in Topology, 2nd edn. Springer, New York (1978) 30. Yi, G.: Continuous extension of preferences. J. Math. Econom. 22, 547–555 (1993)

Chapter 4

On Quasi-uniformities, Function Spaces and Atoms: Remarks and Some Questions Angelika Kontolatou, Kyriakos Papadopoulos, and John Stabakis

Abstract Let F (X, Y ) be the function space that consists of all functions from X to Y , where X and Y are arbitrary non-empty sets. Assume that A is a transitive atom in the lattice of quasi-uniformities on Y , equipped with the partial order relation of inclusion. This type of quasi-uniformities is known from the work of E. P de Jager and H. P. Künzi. The aim of this article is to present remarks and questions on quasiuniformities on function spaces F (X, Y ), that are generated by atoms. We observe that, although this type of quasi-uniformities can be easily defined, the corresponding topologies do not behave well with respect to the well-known exponential laws. These quasi-uniformities will not necessarily be atoms, thus we state the following problem: how can one define a correspondence between the atoms of the lattice of quasi-uniformities, and those atoms that are in the lattice of all quasi-uniformities on Y , in the function space F (X, Y )? Keywords Quasi-uniformities · Function spaces · Atoms [2000] Primary 54X10 · 58Y30 · Secondary 55Z10

4.1 Introduction In this section, we present some fundamental definitions and results about quasiuniformities and function spaces, that are important to the rest of the paper. This brief exposition is based on material found in [3]. In the words of H.-P. Kunzi, quasiuniformities represent a generalization of metric spaces and partial orders, which allows the unification of common features of these theories (see [14]). Our Chapter

A. Kontolatou · K. Papadopoulos (B) · J. Stabakis Department of Mathematics, University of Patras, Patras, Greece e-mail: [email protected] Department of Mathematics, Kuwait University, Kuwait City, Kuwait © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_4

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focuses on the results of [4] on atoms, but there are recent results (see, for example, [10–13]) which show that the topic has been extended to plenty and different research directions. Definition 1 A quasi-uniformity on a set X is a filter U , on X × X , such that (U1) If U ∈ U , then U ⊇ Δ, where Δ is the diagonal of U . (U2) If U ∈ U , then there exists V ∈ U , such that V ◦ V ⊆ U . The pair (X, U ) is called a quasi-uniform space and the members of U are called entourages. Definition 2 A non-zero subfamily B, of a quasi-uniformity U , is a base for U , on X , if and only if every member of U contains a member of B. It can be easily shown, from the definition of quasi-uniformity, that B is a base for U , if and only if (U1) and (U2) are satisfied, plus the property (F2) of filters, which states that, for every two elements which belong to a filter (B in our case), their intersection belongs to the filter, too. Definition 3 A collection S of subsets of X × X is a subbase for a (quasi-)uniformity U , on X , if and only if the family of the finite intersections of the members of S is a base for U . It can be also easily shown, again from the definition of quasi-uniformity, that S is a subbase for U , if and only if (U1) and (U2) are satisfied.  From now  on the set of all quasi-uniformities on X will be denoted by q(X ), and q(X ), ⊆ will be considered a complete lattice (see, for example, [3]).

4.2 Quasi-Uniformities on Function Spaces The authors of articles [5–9] have studied quasi-uniformities on function spaces in depth. We can define a (quasi-)uniformity on a function space F (X, Y ), in the following way: Definition 4 Let X be a set and (Y, V ) a (quasi-)uniform space. Then S = {(S, V ) : S ⊆ P(X ), V ∈ V } = {( f, g) ∈ F (X, Y ) × F (X, Y ) : ( f (x), g(x)) ∈ V, ∀x ∈ S } is a subbase for a (quasi-)uniformity, on F (X, Y ), called the (quasi-)uniformity of the (quasi-)uniform convergence, on the members of S.

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Definition 5 Let X be a set, let (Y, V ) be a quasi-uniform space and let also B = {B : B ⊆ X } ⊆ P(X ) be a cover for X . Then, the quasi-uniformity q B in the members of the cover B, of X, has as a subbase the family: S = {(B, V ) : B ∈ B, V ∈ V } = {( f, g) ∈ F (X, Y ) × F (X, Y ) : ( f (x), g(x)) ∈ V, ∀x ∈ B} Definition 6 Given arbitrary spaces Y and Z let Ct (Y, Z ) denote the set C(Y, Z ) of continuous maps from Y to Z equipped with some topology t. The topology t is said to be splitting on C(Y, Z ) if, for every space X , the continuity of a function f : X × Y → Z implies that its adjoint map fˆ : X → Ct (Y, Z ) is well-defined, where fˆ(x)(y) = f (x, y) for all x and y. In other words, t is a splitting topology, if the exponential injection E X Y Z : C(X × Y, Z ) → C(X, Ct (Y, Z )), where E X Y Z ( f ) = fˆ, is well-defined (see, for example, [2]). If for every space X , the continuity of fˆ : X → Ct (Y, Z ) implies the continuity of f : X × Y → Z , then t is called jointly continuous (or admissible) on C(Y, Z ). Equivalently, t is jointly continuous on C(Y, Z ) if the evaluation map e : Ct (Y, Z ) × Y → Z is continuous, where e(g, y) = g(y). A splitting-jointly continuous topology, on C(Y, Z ), is both the greatest splitting and the coarsest jointly continuous topology on C(Y, Z ) (see, for example, [1]). The exponential objects of the category Top are exactly the core compact topological spaces.

4.3 Quasi-Uniformities on Function Spaces Generated by Atoms In [4], de Jager and Künzi describe atoms, anti-atoms and complements, in the lattice of quasi-uniformities. In this section, we will construct quasi-uniformities on function spaces generated by atoms and we will examine their topologies. Definition 7 Two comparable, distinct quasi-uniformities on a set X , for which there does not exist a quasi-uniformity strictly in between, are called adjacent or neighbours. Two important special cases of adjacent quasi-uniformities are the upper neighbours of I , the indiscrete quasi-uniformity X × X, which will be called atoms, and the lower-neighbours of D, the discrete quasi-uniformity fil{Δ}, which are called  anti-atoms of the lattice q(X ), ⊆ . In Proposition  1., of [4],the authors give the following characterization for a transitive atom of q(X ), ⊆ .

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  Proposition 1 Let S A = (X − A) × X ∪ (X × A) be a preorder, and A =fil{S A } be the filter generated by S A (where X is a set and A a subset of X ). Then, A is a transitive atom of (q(X ), ⊆). We define a quasi-uniformity, on a function space, which is generated by an atom, as follows:   Definition 8 If A is an atom of q(Y ), ⊆ , then the quasi-uniformity on F (X, Y ), generated by the subbase S = {(X, V ) : V ∈ A } = {( f, g) ∈ F (X, Y ) × F (X, Y ) : ( f (x), g(x)) ∈ V, ∀ x ∈ X } is called the quasi-uniformity of the atom quasi-uniform convergence, and will be denoted by qA . We generalize the above definition, by the following proposition.   Proposition 2 Let A be an atom of q(Y ), ⊆ . Then, S = {( f, g) ∈ F (X, Y ) × F (X, Y ) : f −1 (A) ⊆ g −1 (A)} will be a subbase for qA . Proof A subbase for qA is of the form S = {(X, S A )}   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : ( f (x), g(x)) ∈ S A , ∀ x ∈ X    = ( f, g) ∈ F (X, Y ) × F (X, Y ) : f (x), g(x) ∈ ((Y − A) × Y )  ∪ (Y × A), ∀ x ∈ X    = ( f, g) ∈ F (X, Y ) × F (X, Y ) : f (x) ∈ / A ∨ g(x) ∈ A  ∀x ∈ X   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : ¬ x ∈ f −1 (A)   ∨ x ∈ g −1 (A) , ∀ x ∈ X   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : x ∈ f −1 (A) →   x ∈ g −1 (A) , ∀ x ∈ X   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : f −1 (A) ⊆ g −1 (A)

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For finding the induced topology TqA of qA , we consider the subbasic neighbourhoods of a function f , of F (X, Y ), and we get the following subbase: (X, S A )( f ) = {g ∈ F (X, Y ) : ( f, g) ∈ (X, S A )} 

= g ∈ F (X, Y ) : f −1 (A) ⊆ g −1 (A) Example 1 We consider C (X, S), the space of all continuous functions, X V , from a set X to the Sierpinski space S, whose underline set is {0, 1} and the topology is {∅, {1}, {0, 1}}; 1, x ∈ V X v (x) = 0, otherwise. For, say, A = {1}, the subbasic neighbourhoods for a function X V , in C (X, S), will be

 (X, S A )(X V ) = X U ∈ C (X, S) : X V−1 (1) ⊆ X U−1 (A) = {X U ∈ C (X, S) : V ⊆ U } This topology is not a splitting one. Additionally, it is contained in the Isbell topology, which is the finest splitting topology on C (X, S ). So, there exists H ∈ Ω(X ), such that X V ∈ (H, {1}) ⊆ {xU ∈ C (X, S ) : V ⊆ U }. Thus, V ∈ H and ∀ U ∈ H , U ⊇ V . So, H = {U ∈ O(X ) : U ⊇ V }, and consequently, V is compact. Remark 1 The set A is open in the topology of the atom A , namely TA . Indeed S A (x) = (((X − A) × X ) ∪ (X × A)) (x) = {y ∈ X : (x, y) ∈ ((X − A) × X ) ∪ (X × A)} = {y ∈ X : y ∈ A} =A Thus, A is a neighbourhood for every point which belongs to it, and so A is open in TA . Remark 2 If (X, T ) is a topological space, and A ∈ T , then obviously TA ⊆ T , since A ∈ TA , and since A is subbasic-open in TA . Remark 3 It is known (see, for example, [3]) that if (X, T ) is a topological space, then there always exists a quasi-uniformity U , on X, such that T (U ) = T. This quasi-uniformity is called Pervin’s quasi-uniformity. In addition, the subbase S = {S A } is such that T (U ) = T . We remark that if (X, T ) is a topological space, then Pervin’s quasi-uniformity is the supremum of all atoms, that is qPervin =

A

A∈T

Furthermore, if X = ∅ and q =

 A⊆X

A , then Tq is the discrete topology.

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Proposition 3 Let X be a non-empty set, and let (Y, T ) be a topological space. Let P also be Pervin’s quasi-uniformity, compatible with T . Then, the quasi-uniformity q p , on F (X, Y ), has as subbase the family: S = {( f, g) ∈ F (X, Y ) × F (X, Y ) : f −1 (A) ⊆ g −1 (A), ∀A ∈ T } Proof Since P = sup A∈T A, it immediately follows that q p = sup qA . We will now define the quasi-uniformity of the atom quasi-uniform convergence, in the members of the cover B of X , which will be denoted by qAB .   Proposition 4 Let A be an atom of q(Y ), ⊆ and let B = {B : B ⊆ X } ∈ P(X, Y ) be a cover for X. Then, S = {( f, g) ∈ F (X, Y ) × F (X, Y ) : B ∩ f −1 (A) ⊆ g −1 (A)} is a subbase for qAB . Proof A subbase for qAB is of the form   S = (B, S A ) : B ∈ B   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : ( f (x), g(x)) ∈ S A , ∀x ∈ B     = ( f, g) ∈ F (X, Y ) × F (X, Y ) : f (x), g(x) ∈ (Y − A) × Y  ∪ (Y × A), ∀x ∈ B   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : f (x) ∈ / A ∨   g(x) ∈ A , ∀x ∈ B   = ( f, g) ∈ F (X, Y ) × F (X, Y ) : x ∈ f −1 (A)]∨   x ∈ g −1 (A) , ∀x ∈ B = ( f, g) ∈ F (X, Y ) × F (X, Y ) : x ∈ B →   x ∈ f −1 (A) ∨ x ∈ g −1 (A) = ( f, g) ∈ F (X, Y ) × F (X, Y ) : x ∈ B∩   f −1 (A) ⇒ x ∈ g −1 (A) 

= ( f, g) ∈ F (X, Y ) × F (X, Y ) : B ∩ f −1 (A) ⊆ g −1 (A)

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Our next step is to find the induced topology of qAB , namely TqBA . So, we consider the subbasic neighbourhoods of a function f , of F (X, Y ), and we get the following subbase for such a topology: (B, S A )( f ) = {g ∈ F (X, Y ) : ( f, g) ∈ (B, S A )}

 = g ∈ F (X, Y ) : B ∩ f −1 (A) ⊆ g −1 (A) We will compare quasi-uniformities that are induced by atoms, by introducing a comparison of their covers on X . Definition 9 If B1 and B2 are covers for a set X , and if ∀ B1 ∈ B1 ∃ B2 ∈ B2 , such that B2 ⊆ B1 , then B1 is said to be finer than B2 . Proposition 5 If B1 and B2 are covers for a set X, such that B1 is finer than B2 , then: qAB1 ⊆ qAB2 Proof Indeed, (B1 , S A ) ∈ qAB1 ⇔ ( f, g) ∈ {F (X, Y ) × F (X, Y ) : B1 ∩ f −1 (A) ⊆ g −1 (A)}. But since B1 is finer than B2 , then there exists a B2 ∈ B2 , such that B2 ⊆ B1 . So, B2 ∩ f −1 (A) ⊆ B1 ∩ f −1 (A) ⊆ g −1 (A), which implies that B2 ∩ f −1 (A) ⊆ g −1 (A). Thus, (B2 , S A ) ∈ qAB2 , and it follows that qAB1 ⊆ qAB2 . Corollary 1 If qAB1 ⊆ qAB2 , then Tq B1 ⊆ Tq B2 . A

A

The proof of Corollary 1 is similar to the proof of Proposition 5. Example 2 Let B1 = {X }. Then S = {(X, S A )}

 = ( f, g) ∈ F (X, Y ) × F (X, Y ) : X ∩ f −1 (A) ⊆ g −1 (A) 

= ( f, g) ∈ F (X, Y ) × F (X, Y ) : f −1 (A) ⊆ g −1 (A) . Let also B2 = {{x} : x ∈ X }. Then S = {(x, S A )}

 = ( f, g) ∈ F (X, Y ) × F (X, Y ) : {x} ∩ f −1 (A) ⊆ g −1 (A) = {( f, g) ∈ F (X, Y ) × F (X, Y ) : f (x) ∈ A ⇒ g(x) ∈ A} . Let, finally, B3 = {K : K ⊆ X is compact}. Then S = {(K , S A )}

 = ( f, g) ∈ F (X, Y ) × F (X, Y ) : K ∩ f −1 (A) ⊆ g −1 (A) . Then, qAB1 ⊆ qAB2 ⊆ qAB3 , and thus Tq B1 ⊆ Tq B2 ⊆ Tq B3 . A

A

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Remark 4 Let A be a transitive atom of Y . Then, A = filt SA . Question: what is the conjugate quasi-uniformity of A and of A −1 ? Answer: we note that  −1 −1 = (X \ A ) × X ∪ X × A SA = X × (X \ A ) ∪ X × A = A × X ∪ X × (X \ A ). That is,

A −1 = filt(S X \A ).

So, A −1 is also a transitive atom of Y . Obviously q A∨A−1 = q A ∨ q A−1 , where v is the supremum of two quasi-uniformities.

  Proposition 6 Let A = filt(S A ), where A ⊂ Y is a transitive atom of q(Y ), ⊆ . Then   S = ( f, g) ∈ F (X, Y ) × F (X, Y ) : f −1 (A) = g −1 (A) is a subbase for the uniformity q A∨A−1 .

Proof It is enough to observe that (X, S −1 ) = {( f, g) ∈ F (X, Y ) : f −1 (A) ⊃ g −1 (A)}. Remark 5 If we consider again C(X, S) and A = {1}, we remark that the quasiuniformity q A∨A−1 is precisely the discrete one, because   (X, S A ) = (x V , xU ) ∈ C(X, S) × C(X, S) : x V−1 (1) ⊆ xU−1 (1)   = (x V , xU ) : V ⊆ U and

  (X, S A ) = (x V , xU ) ∈ C(X, S) × C(X, S) : U ⊆ V .

Thus   (X, S A ) ∩ (X, S −1 A ) = (x V , xU ) ∈ C(X, S) × C(X, S) : U = V   = (x V , xU ) : x V ∈ C(X, S) = ΔC(X,S) , where ΔC(X,S) is the diagonal of C(X, S).

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  Proposition 7 Let f : X → Y be a function. If A is a transitive atom on q(Y ), ⊆ , then the initial quasi-uniformity on X is the one generated by filt(S f −1 (A) ).   In other words, the initial uniformity on X is also an atom on q(Y ), ⊆ . Proof If A ⊆ Y , then   f 2−1 (S A ) = f 2−1 (Y \ A) × Y ∪ Y × A   = X \ f −1 (A) × X ∪ X × f −1 (A) = S f −1 (A) .

4.4 Open Problems   1. Let A be a transitive atom in q(Y ), ⊆ . How can one define an atom in     q F (X, Y ) , ⊆ , that is generated by A ?   If we denote by α(Y ) all atoms of q(Y ), ⊆ , this question can be restated as follows: Does there exist a map   f : α(Y ) → α F (X, Y ) ? 2. Does there exist a map   f : α(X × Y ) → α F (X, Y ) ? 3. Does there exist a map   f : α F (X, Y ) → α(X )? 4. Does there exist a map   f : α F (X, Y ) → α(Y )? 5. What are the corresponding answers, to the questions above, if we consider an anti-atom in (q(Y ), ⊆)?

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References 1. Arens, R., Dugundgi, J.: Topologies for function spaces. Pacific J. Math. 1, 5–31 (1951) 2. Dugundgi, J.: Topology. Allyn and Bacon, Boston, MA, USA (1966) 3. Fletcher, P., Lindgren, W.F.: Quasi-Uniform Spaces. In: Lecture Notes in Pure and Applied Mathematics, vol. 77 (1982) 4. de Jager, E.P., Künzi, H.-P.: Atoms, anti-atoms and complements in the lattice of quasiuniformities. Topol. Its Appl. 153, 3140–3156 (2006) 5. Papadopoulos, B.K.: The Topology of Quasi Uniform Convergence on the Set of Continuous Maps, pp. 381–390. III Convegno Nationale Di Toopologia, Trieste, Supplement Di Rendiconti Del Circolo Matematico Di Palermo (1988) 6. Papadopoulos, B.K.: (Quasi-)Uniformities on the set of bounded maps. Int. J. Math. Math. Sci. 17(4), 693–696 (1994) 7. Papadopoulos, B.K.: A note on the paper ‘Quasi uniform convergence on function spaces’. Quest. Answ. Gen. Topol. 13, 55–56 (1995) 8. Georgiou, D.N., Papadopoulos, B.K.: Convergence and topologies on function spaces. Panam. J. Math. 5(1), 101–119 (1995) 9. Georgiou, D.N., Papadopoulos, B.K.: A note on the finest splitting topology, questions and answers in general. Topology 13(2), 137–144 (1997) 10. Andrikopoulos, A., Gounaridis, I.: A non-symmetric completion for quasi- uniform spaces. In: International Conference on Topology and Its Applications, 7–11 July 2018 11. Sanchez, I., Sanchis, M.: Quasi-uniformities and quotients of paratopological groups. Filomat 31(6), 1721–1728 (2017) 12. Sanguly, S., Dutta, K., Sen, R.: A note on quasi-uniformity and quasi-uniform convergence on function space. Carpathian J. Math. 24(1), 46–55 (2008) 13. Gaba, Y.U.: Quasi-Uniform Type Spaces. arXiv:1903.06582 14. Kunzi, H.P.A.: Quasi-uniform spaces. In: Encyclopedia of General Topology, pp. 266–270 (2003)

Chapter 5

Some Cardinal Estimations via the Inclusion-Exclusion Principle in Finite T0 Topological Spaces James F. Peters and Irakli J. Dochviri

Abstract The Inclusion-Exclusion formula and interval arithmetics are used to obtain interval estimations of cardinal numbers of certain basic sets in finite topological spaces satisfying Kolmogorov’s T0 separation axiom. Keywords Finite sets · Cardinal number · Interval mathematics · Inclusion-exclusion property

5.1 Introduction In the point-set topology, trends during the last five decades are connected with investigations of infinite cardinal functions of topological spaces. However, many combinatorial properties of finite topological spaces are in shadow. It is well-known that finite sets play a very important role in the analytical description of real-world problems. The most famous characterizations of finite sets can be expressed via the inclusion-exclusion principle, Euler and Venn diagrams, and they all are mutually connected with each other. The inclusion-exclusion principle remains a very effective method for different kinds of sets to count their cardinal numbers (see, e.g., [2–14, 16, 17]). Here we are motivated by the naturally raised problem of evaluation of cardinality of closure and interior of given sets in the finite topological spaces satisfying T0 -separation axiom (also known as A. Kolmogorov’s axiom) based on the inclusion-exclusion principle. J. F. Peters Department of Electrical and Computer Engineering, University of Manitoba, 75A Chancellor’s Circle, Winnipeg, MB R3T 5V6, Canada e-mail: [email protected] Department of Mathematics, Adiyaman University, 02040 Adiyaman, Turkey I. J. Dochviri (B) Department of Mathematics, Caucasus International University, 73 Chargali str, 0141 Tbilisi, Georgia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_5

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5.2 Preliminaries In 1966, R. Moore developed interval mathematics for computational problems, where the parameters of investigating models are uncertain and we are only able to describe parameters by the closed interval estimations [15]. As we observe below, this approach is suitable for finite cardinal numbers (i.e., numbers of elements of a set), too. A closed interval of the reals is denoted by [a1 , a2 ] = {x ∈ R : a1 ≤ x ≤ a2 }, where a1 , a2 , b1 , b2 , x ∈ R. Due to R. Moore’s interval arithmetic, we have the following operations on closed intervals: (1) (2) (3) (4)

[a1 , a2 ] + [b1 , b2 ] = [a1 + b1 , a2 + b2 ]; [a1 , a2 ] − [b1 , b2 ] = [a1 − b2 , a2 − b1 ]; [a1 , a2 ] × [b1 , b2 ] = [min P, max P], where P = {a1 b1 , a1 b2 , a2 b1 , a2 b2 }; 1 ,a2 ] = [a1 , a2 ] × [ b12 , b11 ]. If 0 ∈ / [b1 , b2 ], then [a [b1 ,b2 ]

It should be specially noticed that any real number k is identified with an interval [k, k]. Moreover, if a1 and b1 are non-negative real numbers, then interval multiplication (3) should be changed in the following way: [a1 , a2 ] × [b1 , b2 ] = [a1 b1 , a2 b2 ]. Also, if a1 ≥ 0 and b1 > 0, then instead the rule (4) we use its sim[a1 ,a2 ] = [ ab21 , ab21 ]. Obviously, if we know two possible cardinal estimaplified form [b 1 ,b2 ] tions of a set A in the closed interval form as following car d(A) ∈ [a1 , a2 ] and car d(A) ∈ [b1 , b2 ] with [a1 , a2 ] ∩ [b1 , b2 ] = ∅, then we should declare car d(A) ∈ [max{a1 , b1 }, min{a2 , b2 }], where the numbers a1 , a2 and b1 , b2 are non-negative integers. Below, we deal only with estimations for cardinal numbers with nonnegative integral bounds. The following interval observations are easily verified: If A ⊂ B are subsets of a set X with car d(X ) = n and car d(A) ∈ [a1 , a2 ], where a2 < n. Then car d(B) ∈ [a1 + 1, n − 1]. Also, if X = A ∪ B is a finite set with car d(X ) ∈ [m, n], but car d(A) ∈ [a1 , a2 ] and car d(B) ∈ [b1 , b2 ]. Then car d(A ∩ B) ∈ [a1 + b1 − n, a2 + b2 − m]. If A1 ∩ A2 = ∅ and we know the exact values of car d(A1 ) and car d(A2 ), then car d(A1 ∩ A2 ) ∈ [1, min{car d(A1 ), car d(A2 )}] and car d(A1 ∪ A2 ) ∈ [max {car d(A1 ), car d(A2 )}, car d(A1 ) + car d(A2 ) − 1]. The sets of all integers and all natural numbers are denoted by Z = {0, ±1, ±2, . . .} and N = {0, 1, 2, . . .}, respectively. Moreover, we have to use integer-valued operations q = max{m ∈ Z|m ≤ q} and q = min{n ∈ Z|n ≥ q}, which are well-defined for any rational number q. It can be easily verified that if A, B and C are finite sets such that C = A × B, car d(C) ∈ [c1 , c2 ] and car d(A) ∈ [a1 , a2 ], then car d(B) ∈ [ ac12 , ac21 ]. For topological spaces, we use the following notions from [1]. If O ⊂ X is an open subset of a topological space (X, τ ), then we will write O ∈ τ . The complement of an open set is called a closed set. The collection of all closed subsets of (X, τ ) we denote by co(τ ). Also, in a topological space (X, τ ), denote by cl(A) the closure (resp. int(A) the interior of) A ⊂ X , which is the smallest closed (resp. the largest open) set containing (resp. contained in) a set A.

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Topological closure and interior operators are explicitly related each other as follows: cl(A) = X \ (int(X \ A)) and int(A) = X \ (cl(X \ A)). The family of dense subsets of (X, τ ) denote by D(X ), i.e. D(X ) = {G ⊂ X |cl(G) = X }. Recall that a topological space (X, τ ) is T1 space if and only if {x} = cl({x}) for every x ∈ X (i.e. any point is closed). Classification of topological spaces via the separation axioms provides us with a very interesting type of T0 spaces. A space (X, τ ) is called T0 , if for every pair of distinct points, at least one of them has a neighborhood not containing the other. It should be emphasized that any T0 space is a T1 space, but the converse does not hold.

5.3 Main Results Below we will use the following simple observation in T0 topological spaces: car d(cl({x})) ∈ [1, n], for any x ∈ X . Theorem 1 Let A = {a1 , a2 } ∈ D(X ) be a subset of a T0 topological space (X, τ ) with car d(X ) = n ∈ [2, k1 + k2 ] where cl({ai }) ∈ [1, ki ], for i ∈ {1, 2}. Then car d (cl({a1 }) ∩ cl({a2 })) ∈ [0, k1 + k2 − n]. Proof Note that for the set A = {a1 , a2 }, we can write its closure as follows: cl(A) = cl({a1 }) ∪ cl({a2 })). In view of A ∈ D(X ), we can write n = car d Hence, (cl(A)) = car d(cl({a1 })) + car d(cl({a2 })) − car d(cl({a1 }) ∩ cl({a2 })). car d(cl({a1 }) ∩ cl({a2 }) = [1, k1 ] + [1, k2 ] − n = [2, k1 + k2 ] − [n, n]=[2 − n, k1 + k2 − n] = [0, k1 + k2 − n].   Theorem 2 Let A = {a1 , a2 } be a subset of a T0 topological space (X, τ ) with car d(X ) = n, where cl({ai }) ∈ [1, ki ], for i ∈ {1, 2}. If cl({a1 }) ∩ cl({a2 }) = ∅, then car d(cl(A)) ∈ [2, min{n, (k1 + k2 )}]. Proof Using two-term inclusion-exclusion formula while k1 + k2 ≤ n, we get car d (cl({a1 , a2 })) = car d(cl({a1 })) + car d(cl({a2 })) − car d(cl({a1 }) ∩ car d(cl({a2 })) = car d(cl({a1 })) + car d(cl({a2 })) = [1, k1 ] + [1, k2 ] = [2, k1 + k2 ].   Theorem 3 Let A = {a1 , a2 } be a subset of a T0 topological space (X, τ ) with car d(X ) = n, where cl({a1 }) ∈ [1, k1 ]. If cl({a2 }) ⊆ cl({a1 }), then car d(cl(A)) ∈ [1, k1 ] with k1 ≤ n. Proof From the condition cl({a2 }) ⊆ cl({a1 }), it follows that car d(cl({a1 }) ∩ car d(cl({a2 })) = car d(cl({a2 })). Hence, n ≥ car d(cl({a1 , a2 })) = car d(cl({a1 })) + car d(cl({a2 })) − car d(cl({a1 }) ∩ car d(cl({a2 })) = car d(cl({a1 })) + car d(cl   ({a2 })) − car d(cl({a2 })) = car d(cl({a1 })). Theorem 4 Let A = {a1 , a2 } ∈ D(X ) be a subset of a T0 topological space (X, τ ) with car d(X ) = n. If cl({a2 }) ⊆ cl({a1 }), then {a2 } ∈ D(X ).

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Proof It is obvious that n = car d(cl(A)) = car d(cl({a1 })) + car d(cl({a2 })) − car d(cl({a1 }) ∩ (cl({a2 })) = car d(cl({a1 })) + car d(cl({a2 })) − car d(cl({a2 })) =   car d(cl({a1 })). So, we get n = car d(cl({a1 })), i.e. {a1 } ∈ D(X ). Moreover, a topological space (X, τ ) is called extremally disconnected (for brevity, we use the abbreviation E.D.), if cl(O) ∈ τ \ {∅}, for any O ∈ τ \ {∅}. It can easily be verified that (X, τ ) is E.D. topological space if and only if cl(O1 ) ∩ cl(O2 ) = ∅, for any pair of disjoint sets O1 , O2 ∈ τ \ {∅}. Theorem 5 Let a topological space (X, τ ) be E.D. with car d(X ) = n. If X = O1 ∪ O2 , where O1 ∈ τ \ {∅} and O2 ∈ τ \ {∅} are disjoint sets, then car d(cl(O1 )) + car d(cl(O2 )) = n. Proof By virtue E.D. of (X, τ ), we get car d(cl(O1 ) ∩ cl(O2 )) = 0. Hence, n =   car d(X ) = car d(O1 ∪ O2 ) = car d(cl(O1 )) + car d(cl(O2 )). Theorem 6 Let A = {a1 , a2 } be a subset of a topological space (X, τ ), such that car d(int(A)) ∈ {0, 1} then int({a1 }) = int({a2 }) = ∅. Proof Note that if car d(int(A)) = 0, then by virtue of int({a1 }) ∪ int({a2 }) ⊂ int({a1 , a2 }), we get car d(int({a1 })) + car d(int({a2 })) = 0. Therefore car d (int({a1 })) = car d(int({a2 })) = 0. If car d(int(A)) = 1, then the inequality car d(int({a1 })) + car d(int({a2 })) < 1 holds. This means that car d(int({a1 })) + car d(int({a2 })) = 0. Hence car d   (int({a1 })) = car d(int({a2 })) = 0. Theorem 7 Let A = {a1 , a2 } be a subset of a topological space (X, τ ) such that car d(int(A)) = 2, then car d(int{a1 }) ∈ {0, 1} and car d(int{a2 }) ∈ {0, 1}. Proof The inequality car d(int({a1 })) + car d(int({a2 })) < 2, implies that car d (int({a1 })) + car d(int({a2 })) ∈ {0, 1}. Hence, we deal with the following independent three cases: car d(int({a1 })) = car d(int({a2 })) = 0, i.e. int({a1 }) and int ({a2 }) are empty sets ; car d(int({a1 })) = 1 and car d(int({a2 })) = 0, i.e. {a1 } ∈ τ \ {∅} and int({a2 }) is an empty set; car d(int ({a1 })) = 0 and car d(int({a2 })) = 1,   i.e. int({a1 }) is an empty set and {a2 } is an open set. Now we present more general combinatorial theorems involving topological conditions for finite T0 spaces. Theorem 8 Let A = {a1 , a2 , . . . , am } be a subset of a T0 topological space (X, τ ) with car d(X ) = n and m ∈ [1, n2 ]. If min{car d(F)|F ∈ co(τ ) \ {∅}} = 2 and cl({ai }) ∩ cl({a j }) = ∅, for i, j ∈ {1, 2, . . . , m} and i = j, then car d(cl(A)) ∈ [2m, n]. Proof Since cl(A) = cl({a1 }) ∪ cl({a2 }) ∪ · · · ∪ cl({am }), the following inequalities hold: n ≥ car d(cl(A)) = car d(cl({a1 })) + car d(cl({a2 })) + · · · + car d(cl({am })) ≥ 2m.  

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Theorem 9 Let A = {x1 , x2 , . . . , x p } be a subset of a T0 topological space (X, τ ) with car d(X ) = n, p ∈ [ n2 , n] and min{car d(F)|F ∈ co(τ ) \ {∅}} = 2. If car d (cl(x)) ∈ [2, k x ], for any point x ∈ (X \ A) and cl({xi }) ∩ cl({x j }) = ∅, for i, j ∈ { p + 1, p + 2, . . . , n}, i = j, then  car d(int(A)) ∈ [0, 2 p − n], for n ≤ x∈(X \A) k x and car d(int(A)) ∈ [n −

 x∈(X \A)

k x , 2 p − n], for

 x∈(X \A)

k x < n.

Proof Assume that X = {x1 , x2 , . . . , xn } and A = {x1 , x2 , . . . , x p }. It is known that int(A) = X \ cl(X \ A). Since car d(X \ A) = n − p by Theorem 8, we can write car d(cl(X \ A)) ∈ [2(n − p), n]. But, taking into account the condition car d(cl(xi )) ∈ [2, ki ], i = p + 1, n, we obtain a better estimation than previously, namely: car d + · · · + car d(cl(xn )) ∈ [2, k p+1 ] (cl(X \ A)) = car d(cl(x p+1 )) + car d(cl(x p+2 ))  n + [2, k p+2 ] + · · · + [2, kn ] = [2(n − p), min{n, i= p+1 ki }]. n It is clear that if n ≤ i= p+1 ki , then car d(cl(X \ A)) ∈ [2(n − p), n]. Hence, n car d(cl(X \ A)) ∈ we get car d(int(A)) ∈ [0, 2 p − n]. If i= p+1 ki < n, then n n [2(n − p), i= p+1 ki ] and we obtain car d(int(A)) ∈ [n − i=  p+1 ki , 2 p − n].  Recall that a set A of a topological space (X, τ ) is called semi-open, if there exists O ∈ τ \{∅} such that O ⊂ A ⊂ cl(O). The complement of a semi-open set is called semi-closed. For the class of all semi-open (resp. semi-closed) subsets of a space (X, τ ), we denote usually as S O(X ) (resp. SC(X )). It can be easily verified that A ∈ S O(X ) if and only if A ⊂ cl(int(A)), but B ∈ SC(X ) if and only if int(cl(B)) ⊂ B. Theorem 10 Let A ∈ S O(X ) be a nonempty subset of a T0 topological space (X, τ ) with car d(X ) = n. Then there exists k ∈ N \ {0} such that car d(A) ∈ [k + 1, 2k − 1], for k ∈ [1, n2 ]. Proof For a set A ∈ S O(X ), we can choose O ∈ τ \{∅} such that O ⊂ A ⊂ cl(O). Hence, car d(O) < car d(A) < car d(cl(O)) ≤ n. Denote by k = car d(O), then it is obvious that k ∈ [1, n − 1]. Hence, car d(A) ∈ [k + 1, n − 1], but by Theorem 8, we can write car d(cl(O)) ∈ [2k, n]. Note that the inequality 2k < n implies k ∈ [1, n2 ]. Collecting our estimations, we get k + 1 ≤ car d(A) < [2k, n], i.e. car d(A) ∈ [k + 1, 2k − 1].  

5.4 Conclusion In our present investigation, we have established several cardinal estimations of certain subsets of finite topological spaces with the T0 separation axiom and involving methods of interval mathematics.

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Acknowledgements We would like to express our sincere gratitude to professor Klaus Dohmen, from the University of Applied Sciences in Mittweida, for sharing his rich experiences toward inclusion-exclusion principle and connecting important references. The first author also thanks Banavar V. Saroja, Anna Di Concilio and Tane Vergili for their many inspirational discussions and insights concerning topics related to those in this paper. We also thank the anonymous reviewers for their useful comments.

References 1. Alexandroff, P., Hopf, H.: Topologie I. Springer, Berlin (1935) 2. Attali, D., Edelsbrunner, H.: Inclusion-exclusion formulas from independent complexes. Discret. Comput. Geom. 37(1), 59–77 (2007) 3. Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39, 546–563 (2009) 4. Boag, A., Gull, E., Cohen, G.: Inclusion-exclusion principle for many-body diagrammatics. Phys. Rev. B 98(11), 115152 (2018) 5. Dochviri, I., Peters, J.F.: Topological sorting of finitely near sets. Math. Comput. Sci. 10(2), 273–277 (2016) 6. Dohmen, K.: Inclusion-exclusion: which terms cancel? Arch. Math. 74(4), 314–316 (2000) 7. Dohmen, K.: An improvement of the inclusion-exclusion principle. Arch. Math. 72(4), 298–303 (1998) 8. Dohmen, K.: Improved inclusion-exclusion identities via closure operators. Discret. Math. Theor. Comput. Sci. 4(1), 61–66 (2000) 9. Dohmen, K.: Improved Bonferroni Inequalities Via Abstract Tubes. Springer, Berlin (2003) 10. Edelsbrunner, H., Ramos, E.A.: Inclusion-exclusion complexes for pseudo-disk collections. Discret. Comput. Geom. 17, 287–306 (1997) 11. Goaoc, X., Matou˘sek, J., Paták, P., Safernová, Z., Tancer, M.: Simplifying Inclusion - Exclusion Formulas. Comb. Probab. Comput. 24(2), 438–456 (2015) 12. Kahn, J., Linial, N., Samorodnitsky, A.: Inclusion-exclusion: exact and approximate. Combinatorica 16(4), 465–477 (1996) 13. Kratky, K.W.: The area of intersection of n equal circular disks. J. Phys. A 11(6), 1017–1024 (1978) 14. Linial, N., Nisan, N.: Approximate inclusion-exclusion. Combinatorica 10(4), 349–365 (1990) 15. Moore, R.E.: Interval Analysis. Prentice-Hall, N.J. (1966) 16. Naiman, D.Q., Wynn, H.P.: Inclusion-exclusion-Bonferroni identities and inequalities for discrete tube like problems via Euler characteristics. Ann. Stat. 20(1), 43–76 (1992) 17. Peters, J.F., Kordzaya, K., Dochviri, I.: Computable proximity of l1 discs on the digital plane, CAMS. Commun. Adv. Math. Sci. 2(3), 213–218 (2019)

Chapter 6

Representations of Preference Relations with Preutility Functions on Metric Spaces Yann Rébillé

Abstract The elaboration of preference relations and their representations take back their source in early economic theory. Classical representations of preferences theorems rely on Debreu-Eilenberg’s theorems through abstract mathematics based on topological properties. We adopt another approach starting from metric spaces instead. We obtain representation theorems of preference relations with bivariate functions. This allows us to handle intransitivities and incomparabilities of the preference relation and also continuity conditions of various strength. Keywords Preference relation · Utility function · Metric space

6.1 Introduction A basic starter for a first course in microeconomics includes indifference curves, utility functions and then develops the graphical analysis and the related optimization techniques. Utility functions provide numerical representations of preferences and then calculus can be performed. The elaboration and reflexion on utility theory can be traced back to early economic theory [18]. Interestingly, the sources of utility representation theory are multidisciplinary (e.g. psychology, cognitive sciences, chemistry, physics, etc.) and can be handled from a purely mathematical point of view (see [5]). One may easily switch from indifference curves to utility functions as soon as preferences are considered as objects of study per se (i.e. binary relations). We may quote G. Debreu (see Sect. 4 p. 403 in [12]): “Although more abstract than the familiar concept of an infinite family of indifference sets in Rl , the concept of a single set G in Rl × Rl is far simpler [...].” For preferences relations that are continuous weak orders the existence of continuous utility functions is guaranteed Y. Rébillé (B) LEMNA, IAE de Nantes, University of Nantes, chemin la Censive du Tertre, BP 52231, Nantes Cedex 3, 44322 Nantes, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_6

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as soon as the underlying topological space is second countable or connected and separable (in [10, 11, 14]). Typically, one encounters the space of commodities Rn+ as a standard framework in microeconomics, which is an instance of connected and separable metric space based on the Euclidean space (see [9]). Still, utility theory relies on sophisticated abstract mathematics (set theory, topology) and various techniques can be used to derive these results (see [4, 21] and references therein). Our motivation is to obtain simple representations of preference relations on metric spaces. However, since we focus only on the metric aspect and not on the topological aspect we will obtain bivariate functions instead of utility functions. Here, for any two elements x, y (e.g. prospects, plans, bundles, ...), x is strictly preferred to y if and only if the overall changes by switching from y to x are beneficial to the agent. In some way, there is a flow (of gains/losses, pleasures/pains, ...) from y to x which is deemed as being positive. Flow functions and their properties will be presented in the sequel. This framework is flexible and allows one to handle incomplete preferences and intransitivities too. Our method is based on a separation result on metric spaces and does not require additional knowledge in abstract mathematics (e.g. topology). Our approach is close to Beardon’s work [1] on the representation of preferences by continuous utility functions on metric spaces. This approach is known as Arrow-Hahn’s approach (see [1] and references therein). Our contribution can be seen as an extension of Shafer’s numerical representation of continuous total binary relations on Rn+ [23]. This extension addresses the general case of metric spaces, with general preferences and under continuity conditions of various strength. In Sect. 6.2, we present the usual setting of preference theory. Section 6.3 recalls Debreu’s theorems. Section 6.4 introduces flow functions. Sections 6.5 and 6.6 provide representation theorems of preferences by preutility functions and utility functions. Section 6.7 gives some applications and interpretations of preutility functions for theoretical economics.

6.2 Notations, Definitions We can refer essentially to [4] for a classical exposition. Let X denote an arbitrary nonempty set and x ∈ X is a generic element. A topology of open sets is denoted by τ . (X, d) denotes a metric space endowed with a distance function d. The topology generated by d is denoted τd or τ when no confusion is possible. X is separable if there exists a countable set Z ⊂ X such that cl(Z ) = X where cl(.) denotes the closure. X is second countable or admits a countable basis if there exists a countable family of open sets such that any open set is a union of these sets. X is connected if the only sets that are open and closed are ∅ and X . A function u : X −→ R is continuous if for all α ∈ R, {x ∈ X : u(x) < α} ∈ τ and {x ∈ X : u(x) > α} ∈ τ .

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A binary relation on X is denoted by  ⊂ X 2 . Then, x  y ⇐⇒ (x, y) ∈  for x, y ∈ X . The dual preference relation is denoted by where x y if and only if y  x. Similarly, ∼ = ∩ , ≺ =  \ ∼ and = \ ∼.  is reflexive if for all x ∈ Y , x  x (denoted (RF)).  is transitive if for all x, y, z ∈ X, x  y, y  z ⇒ x  z (denoted (TR)).  is total if for all x, y ∈ X , x  y or y  x (denoted (TL)). Clearly, (TL) implies (RF), and ≺ =  is equivalent to (TL).  is a preorder if  satisfies (RF), (TR). A total preorder is called a weak order. A reflexive binary relation is called an interval order [4, 15] if for all x, y, z, t ∈ X , x  y and z  t implies x  t or z  y. An order interval satisfies (TL), since for all x, y ∈ X , x  x and y  y by (RF), thus x  y or y  x. Weak orders are interval orders.1 A function u : X −→ R is a utility representation of  if and only if for all x, y ∈ X , x  y ⇐⇒ u(x) ≤ u(y) . A couple of functions (u, v) with u, v : X −→ R and u ≤ v is an interval order representation of  if and only if for all x, y ∈ X , x  y ⇐⇒ u(x) ≤ v(y) . Necessarily, if  admits a utility representation, then  is a weak order. Necessarily, if  admits an interval order representation, then  is an interval order. A function U : X 2 −→ R is a preutility representation or a bivariate representation of  if and only if for all x, y ∈ X , x  y ⇐⇒ U (x, y) ≤ 0 . Clearly, U (x, y) = u(x) − u(y) or U (x, y) = u(x) − v(y) for x, y ∈ X define preutilities. For x ∈ X , let us denote (←, x] = {y : y  x} and [x, →) = {y : x  y}, (←, x) = {y : y ≺ x} and (x, →) = {y : x ≺ y}. The binary relation  satisfies continuity (denoted (CT)) if, for all x ∈ X, [x, →) and (←, x] are closed. A continuous weak order is a weak order that satisfies (CT). Necessarily, if  admits a continuous utility representation, then  satisfies (CT). A binary relation  is closed if  ⊂ X 2 is closed in the product topology. According to the following Remarks, continuous utility or continuous order interval representations implies closedness which in turn implies (CT). Let x, y, z, t ∈ X with x  y and z  t. If x  t then by (TL) we have t  x. By (TR), t ( x)  y. Now, z  t, so by (TR) z  y. This establishes the interval condition.

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Remark 1 Assume  admits an order interval representation given by (u, v). Then,  is closed. Let x0 , y0 ∈ X with x0 y0 . Then, we have u(x0 ) > v(y0 ). Let α ∈ (v(y0 ), u(x0 )). Since u and v are continuous, there exists Ox0 ∈ τ with x0 ∈ Ox0 such that u(x) > α for x ∈ Ox0 and similarly there exists O y0 ∈ τ with y0 ∈ O y0 such that v(y) < α for y ∈ O y0 . Then, u(x) > α > v(y), so x y for (x, y) ∈ Ox0 × O y0 . This establishes that is open in X 2 , so  is closed. Remark 2 Assume  is closed. Then,  satisfies (CT). Clearly, if  = X × X then for all x ∈ X we have [x, →) = X and (←, x] = X . Assume  = X × X . Let x0 ∈ X . If [x0 , →) = X we are done. Otherwise, let y ∈ X such that x0  y. Since  is open, there exists Ox0 , O y ∈ τ with x0 ∈ Ox0 , y ∈ O y such that Ox0 × O y ⊂ . Thus, we have {x0 } × O y ⊂ , that is O y ⊂ X \[x0 , →) so X \[x0 , →) ∈ τ . Symmetrically, if (←, x0 ] = X we are done. Otherwise, let y ∈ X such that y  x0 . Since  is open, there exists O y , Ox0 ∈ τ with y ∈ O y , x0 ∈ Ox0 , such that O y × Ox0 ⊂ . Thus, we have O y × {x0 } ⊂ , that is O y ⊂ X \(←, x0 ] so X \(←, x0 ] ∈ τ . A converse statement can be established for weak orders and interval orders. Property 1 Assume (X, d) is a metric space.2 Let  ⊂ X 2 be a weak order or an interval order. Then, if  satisfies (CT) then  is closed. Proof Let us recall that whenever  is an order interval it satisfies (TL), so ≺ = . Let {xn }n ⊂ X with limn xn = x and {yn }n ⊂ X with limn yn = y such that for all n ∈ N, xn  yn . Let us proceed by way of contradiction. Assume x y. Since (y, →) ∈ τ by (CT), there exists N1 such that for all n ≥ N1 it holds xn y. Similarly, since (←, x) ∈ τ by (CT), there exists N2 such that for all n ≥ N2 it holds x yn . So for N = N1 ∨ N2 we have y ≺ x N  y N ≺ x. Once more, since x ∈ (y N , →) ∈ τ there exists some N3 ∈ N such that for all n ≥ N3 it holds y N ≺ xn . Similarly, since y ∈ (←, x N ) ∈ τ there exists some N4 ∈ N such that for all n ≥ N4 it holds yn ≺ x N . In particular, for N  = N3 ∨ N4 we have, y N ≺ x N  and y N  ≺ x N . Now since, x N   y N  and since  is an order interval we  have y N ≺ x N , a contradiction. So, x  y holds. This proves that  is closed.

6.3 Some Observations 6.3.1 Debreu’s Theorems We may recall Debreu’s theorem [10, 11]. Theorem 1 ([10, 11]) Let X be a second countable topological space and let  ⊂ X 2 be a binary relation. If  is a continuous weak order then  admits a continuous utility representation. Moreover, it is an ordinal representation, i.e. unique up to an increasing transformation. 2

It is sufficient that X be first countable.

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Similarly, Debreu-Eilenberg’s theorem applies to topological spaces that are connected and separable instead of second countable [10, 11, 14, 21]. A similar result holds for interval orders [8, 22]. For metric spaces, separability and second countability are equivalent. For instance, compact metric spaces are second countable (and complete), and also σ -compact spaces.3 Beardon [1] considers metric spaces that are proper, i.e. any closed ball of any radius B[x, r ] = {y ∈ X : d(x, y) ≤ r } is compact. Proper metric spaces are σ -compact (and complete). However, some metric spaces are neither connected, separable nor second countable. For instance, Example 1 Let X = R and τ = 2R . Then, (X, τ ) with the discrete topology admits the metric defined by d(x, y) = 0 if x = y and d(x, y) = 1 if x = y for x, y ∈ R. Then, (X, d) is neither connected, since {x} is both closed and open for any x ∈ X , nor separable since R is uncountable.

6.3.2 Lexicographic Order The following celebrated example of lexicographic preferences ([10, 11], see also p. 5 in [18]) illustrates the problem of obtaining a utility representation even without any continuity property nor topological property under consideration. This problem remains with interval order representations. Example 2 Let X = R2 . Consider the lexicographic order (which is a total preorder and also anti-symmetric4 ), defined on R2 as follows. Let us say x = (x1 , x2 ) precedes y = (y1 , y2 ), denoted x ≤lex y if x1 < y1 or x1 = y1 and x2 ≤ y2 . However, ≤lex does not admit an interval order representation. Assume on the contrary that there exists an interval order representation (u, v) with u, v : R2 → R and u ≤ v. Then, for α ∈ R, we have (α, 0) α} are open in the product topology. W is called a (continuous) flow if it is (continuous) nonnegative, null-diagonal, asymmetric and null-transitive. The asymmetry condition can be stated alternatively as W (x, y) > 0 ⇒ W (y, x) = 0 for all x, y ∈ X . Hence, W is not symmetric as soon as W = 0. Clearly, W is null-transitive whenever it satisfies the usual triangular inequality, i.e. W (x, z) ≤ W (x, y) + W (y, z) for all x, y, z ∈ X . Let us present an immediate use of flows for preference relations. Let W be a nonnegative bivariate function. Define a binary relation W in the following manner, for all x, y ∈ X , x W y ⇐⇒ W (x, y) = 0 . Then, W is closed as soon as W is continuous since W = W −1 ({0}) and {0} is closed. If W is null-diagonal then W is reflexive. For x ∈ X , since we have W (x, x) = 0, thus x W x. If W is asymmetric then W is total. For x, y ∈ X , if x W y, then W (x, y) > 0. Thus, by asymmetry, W (y, x) = 0, so y W x. If W is null-transitive then W is transitive. For x, y, z ∈ X , if x W y and y W z, then W (x, y) = W (y, z) = 0. Thus, by null-transitivity, W (x, z) = 0, so x W z. So, if W is a (continuous) flow then W is a (continuous) weak order. On metrizable7 topological spaces we have the following representation,

6 7

For all x ∈ X , W (x, x)2 = 0, thus W (x, x) = 0. It suffices only that X be pseudometric, since the separation condition is not useful.

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Theorem 2 Assume (X, d) is a metric space. Let  ⊂ X 2 be a closed reflexive binary relation. Then, there exists a continuous function W on X 2 , nonnegative, null-diagonal, such that for all x, y ∈ X , x  y ⇐⇒ W (x, y) = 0 , which is given by W (x, y) = d 2 ((x, y), ) , where d 2 generates the product topology8 on X 2 . By construction, it holds W ≤ d. In particular, if  is total then W is asymmetric, if  is transitive then W is nulltransitive. And, if  is a weak order then W is a flow. Moreover, any positive continuous transformation is convenient, i.e. φ ◦ W represents  with φ : R+ −→ R+ continuous, φ(0) = 0 and φ(w) > 0 if w > 0. Proof From basics of topology, it is known that d 2 is a metric on X 2 and also that the point-to-set distance function is (1-lipschitz) continuous (e.g. see Theorem 4.3 p.185 in [13]). So, provided  = ∅, d 2 ( . , ) is continuous and d 2 ( . , ) is nonnegative. Since  is reflexive  = ∅. Let x, y ∈ X . If x  y, then (x, y) ∈ , so d 2 ((x, y), ) = 0. And, if d 2 ((x, y), ) = 0 then (x, y) ∈  since  is closed. So, W represents . Let x ∈ X . Since  is reflexive it holds that x  x that is (x, x) ∈ . Thus, d 2 ((x, x), ) = 0. So d 2 ( . , ) is null-diagonal. (By construction). Let x, y ∈ X . Take z = x  = y  . Since  is reflexive we have z  z. So, we have W (x, y) = inf{d(x, x  ) + d(y, y  ) : x   y  } ≤ inf{d(x, z) + d(y, z) : z ∈ X } and also, d(x, y) ≤ inf{d(x, z) + d(y, z) : z ∈ X } ≤ d(x, y) since the first ≤ holds by the triangular inequality and the second ≤ holds taking z = x or z = y. / . (In particular). Let x, y ∈ X with d 2 ((x, y), ) > 0. Then, we have that (x, y) ∈ By totality of  we have that (y, x) ∈ , thus d 2 ((y, x), ) = 0. We may operate mutatis mutandis when d 2 ((y, x), ) > 0. This proves that d 2 ( . , ) is asymmetric. It remains to check that d 2 ( . , ) is null-transitive. Let x, y, z ∈ X with d 2 ((x, y),  ) = 0 and d 2 ((y, z), ) = 0. Since  is closed, we have that (x, y) ∈  and (y, z) ∈ . Now by transitivity of  we have (x, z) ∈ , thus d 2 ((x, z), ) = 0. So d 2 ( . , ) is null-transitive. (Moreover). It is immediate to check. 

d 2 is the distance function defined on X 2 by d 2 ((x, y), (x  , y  )) = d(x, x  ) + d(y, y  ) for all (x, y), (x  , y  ) ∈ X 2 and d 2 ((x, y), F) = inf{d 2 ((x, y), (x  , y  )) : (x  , y  ) ∈ F} where F ⊂ X 2 .

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Remark 3 Any binary relation  ⊂ X 2 can be represented by a bivariate function. It suffices to consider the following, for all x, y ∈ X x  y ⇐⇒ 1 X 2 \ (x, y) = 0 , where 1 X 2 \ is the indicator function of the subset X 2 \ . However, since 1 X 2 \ is {0, 1}-valued as soon as  = ∅, X 2 it will hardly be continuous, except if  is both closed and open.9 Let us deal once more with the example of lexicographic preferences. Then, even if ≤lex does not admit a utility representation, it still admits a continuous flow representation. A less trivial representation is provided in Example 4. Example 3 Let X = R2 and τ = 2R . Consider the lexicographic order denoted by ≤lex . Then, ≤lex ⊂ R2 × R2 is closed, since the product topology of discrete topologies is the discrete topology on the product.10 Thus, ≤lex admits a flow representation, where for all x, y ∈ R2 , 2

x ≤lex y ⇐⇒ W (x, y) = 1 X 2 \≤lex (x, y) = 0 . Here, W is continuous. Some results similar to Theorem 1 can be obtained under weaker continuity requirements such as upper or lower semi-continuity of the preference relation through separate continuity. Theorem 3 Assume (X, d) is a metric space. Let  ⊂ X 2 be a reflexive binary relation. Then, if  is lower semi-continuous (LSC), i.e. for all x ∈ X, (←, x] is closed, there exists a function W on X 2 , nonnegative, null-diagonal, with W (., y) continuous for all y ∈ X , such that for all x, y ∈ X , x  y ⇐⇒ W (x, y) = 0 , which is given by the point-to-set distance, W (x, y) = d(x, (←, y]) = inf{d(x, y  ) : y  ∈ X, y   y} . By construction, it holds W ≤ d. In particular, if  is total then W is asymmetric, if  is transitive then W is nulltransitive. And, if  is a weak order then W is a flow. Conversely, if W is a nonnegative, null-diagonal function on X 2 with W (., y) continuous for all y ∈ X then  satisfies (LSC). 9

That is impossible for instance when X is connected and thus X 2 is connected. Here, d 2 ((x1 , x2 ), (y1 , y2 )) < 1 ⇐⇒ d(x1 , y1 ) + d(x2 , y2 )) < 1 ⇐⇒ d(x1 , y1 ) = d(x2 , y2 )) = 0 ⇐⇒ x1 = y1 and x2 = y2 .

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Similarly, if  is upper semi-continuous (USC), i.e. for all x ∈ X, [x, →) is closed, there exists a function W on X 2 , nonnegative, null-diagonal, with W (x, .) continuous for all x ∈ X , such that for all x, y ∈ X , x  y ⇐⇒ W (x, y) = 0 , which is given by the point-to-set distance, W (x, y) = d(x, [y, →)) = inf{d(x, y  ) : y  ∈ X, y  y  } . By construction, it holds W ≤ d. In particular, if  is total then W is asymmetric, if  is transitive then W is nulltransitive. And, if  is a weak order then W is a flow. Conversely, if W is a nonnegative, null-diagonal function on X 2 with W (x, .) continuous for all x ∈ X , then  satisfies (USC). Moreover, any positive continuous transformation is convenient, i.e. φ ◦ W represents  with φ : R+ −→ R+ continuous, φ(0) = 0 and φ(w) > 0 if w > 0. Proof Since  is reflexive for all y ∈ X , it holds that y  y, so (←, y] = ∅ and then d( . , (←, y]) is well defined and so is d( . , (←, . ]) and both are nonnegative. It is known that the point-to-set distance function d(., (←, y]) is continuous given y is fixed. Let x, y ∈ X . If x  y, then x ∈ (←, y] so d(x, (←, y]) = 0. Now, if d(x, (←, y]) = 0 then since (←, y] is closed by (LSC), we have x ∈ (←, y], that is x  y. So, W represents . Let x ∈ X . Since  is reflexive it holds that x  x that is x ∈ (←, x]. Thus, d(x, (←, x]) = 0. So d( . , (←, . ]) is null-diagonal. (By construction). Let x, y ∈ X . Take y  = y, then since y  y holds by reflexivity, we have W (x, y) ≤ d(x, y  ) = d(x, y). (In particular). Let x, y ∈ X with d(x, (←, y]) > 0. Thus, x ∈ / (←, y]. By totality of  it holds that y  x, so d(y, (←, x]) = 0. We may operate mutatis mutandis when d(y, (←, x]) > 0. This proves that d( . , (←, . ]) is asymmetric. Let x, y, z ∈ X with d(x, (←, y]) = 0 and d(y, (←, z]) = 0. Since  satisfies (LSC), we have that x ∈ (←, y] and y ∈ (←, z], that is x  y and y  z. Now by transitivity of  we have x  z, thus d(x, (←, z]) = 0. So d( . , (←, . ]) is null-transitive. (Converse). Assume W is a nonnegative, null-diagonal function on X 2 such that W (., y) is continuous for all y ∈ X . Then,  is a reflexive binary relation. And also, for all y ∈ X we have (←, y] = W (., y)−1 ({0}) which is closed since W (., y) is continuous, so  satisfies (LSC). (Similarly). The proof mimics the former proof mutatis mutandis. (Moreover). It is immediate to check.  We may consider a similar statement under weaker conditions.

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Theorem 4 Assume (X, d) is a metric space. Let  ⊂ X 2 be a reflexive binary relation. Then, there exists a function W on X 2 , nonnegative, null-diagonal, with W ≤ d, such that for all x, y ∈ X , x  y ⇐⇒ W (x, y) = 0 , which is given by  W (x, y) = 1 X 2 \ × d(x, y) =

d(x, y), if x  y 0, otherwise.

In particular, if  is total then W is asymmetric, if  is transitive then W is nulltransitive. And, if  is a weak order then W is a flow. Moreover, any positive transformation is convenient, i.e. φ ◦ W represents  with φ : R+ −→ R+ continuous at 0, φ(0) = 0 and φ(w) > 0 if w > 0. Proof Clearly, W is nonnegative and W ≤ d. Let x ∈ X . By reflexivity, x  x, so W (x, x) = d(x, x) = 0. So W is null-diagonal. Let x, y ∈ X with x  y. Then, W (x, y) = 0. Let x  y. Then by reflexivity it holds x = y, thus d(x, y) > 0 and then W (x, y) = d(x, y) > 0. This establishes that W represents . (In particular). Let x, y ∈ X with W (x, y) > 0. Then, x  y, so by totality y  x and then W (y, x) = 0, so W is asymmetric. Let x, y, z ∈ X with W (x, y) = W (y, z) = 0. Then x  y and y  z. By transitivity x  z, so W (x, z) = 0. (Moreover). It is immediate to check.  Remark 4 Assume (X, d) is a metric space. Let  ⊂ X 2 be a binary relation which is not reflexive. Then, there exists a function W  on X 2 , nonnegative, null-diagonal, with W  ≤ d, such that for all x, y ∈ X with x = y, we have x  y ⇐⇒ W  (x, y) = 0 . Indeed, it suffices to consider the extension  =  ∪  with  = {(x, x) : x ∈ X }. Then,  is reflexive and admits a representation through a nonnegative, null-diagonal function W  on X 2 with W  ≤ d. Example 4 Let X = R2 and τ = τd2 the euclidean topology given by the euclidean distance function d2 on R2 . Consider the lexicographic order denoted by ≤lex . Then, ≤lex admits a flow representation dominated by d2 , where for all x, y ∈ R2 , x ≤lex y ⇐⇒ Wlex (x, y) = 0 , which is given by, Wlex (x, y) = 1(R2 )2 \≤lex × d2 (x, y) =

 (x1 − y1 )2 + (x2 − y2 )2 , if x lex y 0, otherwise.

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We may observe that Wlex is not separately continuous. For instance, consider xn = (−1/n, 1) and x = (0, 1). Then xn ≤lex y = (0, 0) for all n, d2 (xn , x) = 1/n → 0 but x lex y. So, ≤lex does not satisfy (LSC). Similarly, consider yn = (1/n, 0) and y = (0, 0). Then x = (0, 1) ≤lex yn for all n, d2 (yn , y) = 1/n → 0 but x lex y. So, ≤lex does not satisfy (USC). In general, the condition W ≤ d does not guarantee that W is continuous on X 2 (see Theorem 1), nonetheless we can identify regions where W is continuous, and we can eventually factorize W through d. Property 2 Let W be given by Theorem 3. The following statements hold, (i). The restrictions W| and W|X 2 \ are continuous. Moreover, W is continuous on  = {(x, x) : x ∈ X }. (ii). If  is total and closed then W is asymmetric and continuous on (and ). (iii). Assume that for all x, y, x  , y  ∈ X it holds, d(x, y) = d(x  , y  ) ⇒ W (x, y) = W (x  , y  ) , then there exists φ such that W = φ ◦ d with φ(0) = 0 and φ ≤ I d on d(X 2 ). Proof (i). Clearly since W = 0 on , W| is continuous. Similarly, W = d on X 2 \  which is continuous on (X 2 , d 2 ), hence W|X 2 \ is continuous. (Moreover). Let x0 ∈ X . Then, for all x, y ∈ X we have W (x, y) − W (x0 , x0 ) = W (x, y) − 0 ≤ d(x, y). Let (x, y) tend to (x0 , x0 ), we have d 2 ((x, y), (x0 , x0 )) = d(x, x0 ) + d(y, x0 ) → 0. Thus, by the triangular inequality, W (x, y) ≤ d(x, y) ≤ d(x, x0 ) + d(x0 , y) → 0. (ii). Assume  is total and closed. Then, = X 2 \  and is open. Let (x0 , y0 ) ∈ . Since is open, for some  > 0 we have Bd 2 ((x0 , y0 ), ) ⊂ . Now, for any  ∈ (0, ) we have, for all (x, y) ∈ Bd 2 ((x0 , y0 ), ), W (x, y) − W (x0 , y0 ) = d(x, y) − d(x0 , y0 ) ≤ d(x, x0 ) + d(x0 , y0 ) + d(y0 , y) − d(x0 , y0 ) = d 2 ((x, y), (x0 , y0 )) <  . Similarly, we have W (x, y) − W (x0 , y0 ) = d(x, y) − d(x0 , y0 ) ≥ d(x, y) − (d(x0 , x) + d(x, y) + d(y, y0 )) = −d 2 ((x, y), (x0 , y0 )) > − . This establishes that W is continuous on (since W is locally 1-lipschitz on ). (iii). It is immediate to establish.  Example 5 Let (X, d) be a metric space. For α > 0, we may introduce a dependence relation α through ϕα ( p) = ( p − α)+ for p ≥ 0, where for all x, y ∈ X 2 ,

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x α y ⇐⇒ Wα (x, y) = ϕα ◦ d (x, y) = (d(x, y) − α)+ = 0 ⇐⇒ d(x, y) ≤ α .

Clearly, α is reflexive and symmetric. And α is transitive whenever d is an ultrametric, i.e. for all x, y, z ∈ X we have d(x, y) ≤ sup(d(x, z), d(y, z)).

6.5 Representation Theorems with Preutility Functions We may associate a preutility function to any nonnegative asymmetric function in the following manner. Let W be a (continuous) asymmetric function representing , then UW : X 2 −→ R : (x, y) → UW (x, y) = W (x, y) − W (y, x) is a (continuous) preutility function representing . Indeed, if x  y then W (x, y) = 0, so UW (x, y) ≤ 0. Conversely, if UW (x, y) ≤ 0, then W (x, y) > 0 is impossible, otherwise by asymmetry W (y, x) = 0 and then UW (x, y) > 0. So, W (x, y) ≤ 0 and finally x  y. For instance, let  be represented by a (continuous) function u : X −→ R. Then, for all x, y ∈ X , x  y ⇐⇒ (u(x) − u(y))+ = 0 , where x + = max(x, 0) for x ∈ R. So, Wu (x, y) = (u(x) − u(y))+ is a (continuous) flow, and UWu (x, y) = (u(x) − u(y))+ − (u(y) − u(x))+ = u(x) − u(y). We may reformulate Theorem 1 with preutility functions. The next corollary is an extension of Shafer’s result11 Shafer [23] to metric spaces instead of Rn+ . Corollary 1 (Theorem 1 p. 915 in [23]) Assume (X, d) is a metric space. Let  ⊂ X 2 be a closed total binary relation. Then, there exists a continuous preutility function U : X −→ R, such that for all x, y ∈ X , x  y ⇐⇒ U (x, y) ≤ 0 , more over U can be chosen such that: U is null-diagonal, i.e. U (x, x) = 0 for all x ∈ X , and U is asymmetric, i.e. U (x, y) ≤ 0 or U (y, x) ≤ 0 for all x, y ∈ X ; or U is skew-symmetric12, ,13 i.e. 11

Shafer’s original numerical representation of preferences is defined through the distance to the indifference relation, ∼, instead of the distance to  in our Theorem 1. By definition, k(x, y) = m(x, y) if x y and k(x, y) = −m(y, x) if y x with m(x, y) = min{d((x, y), (z, w)) : z ∼ w} where d is an Euclidean metric on Rn+ × Rn+ . 12 If U is skew-symmetric then U (x, x) = −U (x, x) thus U (x, x) = 0 and U (x, y) > 0 ⇒ U (y, x) = −U (x, y) < 0, hence U is null-diagonal and asymmetric. 13 Typically, take X = Rn and consider B a skew-symmetric real matrix, i.e. B T = −B, then define U (x, y) = x T By for x, y ∈ Rn .

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U (x, y) = −U (y, x) for all x, y ∈ X . In particular, if  is a weak order then U is transit-negative, i.e. U (x, y) ≤ 0, U (y, z) ≤ 0 ⇒ U (x, z) ≤ 0 for all x, y, z ∈ X . Conversely, if U is a continuous, null-diagonal and asymmetric function, or a skewsymmetric function, then U defined by, for all x, y ∈ X , x U y ⇐⇒ U (x, y) ≤ 0 , is a closed total binary relation. In particular, if U is transit-negative then U is a weak order. Moreover, any signed continuous transformation is convenient, i.e. φ ◦ W represents  with φ : R −→ R continuous, φ(0) = 0 and w × φ(w) > 0 if w = 0. Proof We may obtain from Theorem 1 a continuous function null-diagonal and asymmetric function W such that for all x, y ∈ X , x  y ⇐⇒ W (x, y) = 0. Let us consider the continuous preutility function UW : X −→ R, defined for all x, y ∈ X by UW (x, y) = W (x, y) − W (y, x) . By construction, UW is continuous and skew-symmetric, thus null-diagonal and asymmetric. Let x, y ∈ X . Assume x  y. Then, W (x, y) = 0 and we have UW (x, y) = − W (y, x) ≤ 0. Assume x  y. Then, W (x, y) > 0 and by asymmetry we have W (y, x) = 0, thus UW (x, y) = W (x, y) − 0 > 0. So UW represents . (In particular). Let x, y, z ∈ X with UW (x, y) ≤ 0 and UW (y, z) ≤ 0. Thus, W (x, y) = 0 and W (y, z) = 0, and by null-transitivity we have W (x, z) = 0. Hence, UW (x, z) ≤ 0. (Converse). Let U be a continuous, null-diagonal, asymmetric (or skew-symmetric) function. We have U = U −1 ((−∞, 0]) which is closed in X 2 since U is continuous. Let x ∈ X . Then, U (x, x) = 0 by null-diagonality, so x U x. This proves that U is reflexive. Let x, y ∈ X . Then, either U (x, y) ≤ 0 or U (y, x) ≤ 0 holds by asymmetry, thus x U y or y U x. This proves that U is total. (In particular). Let x, y, z ∈ X with x U y, y U z. Then, U (x, y) ≤ 0 and U (y, z) ≤ 0. By transit-negativity, we have U (x, z) ≤ 0, thus x U z. This proves that U is transitive. (Moreover). It is immediate to check.  Clearly, if U is respectively null-diagonal, asymmetric, transit-negative then W = U + is respectively null-diagonal, asymmetric, transit-negative. For instance, let  be represented by a continuous interval order (u, v) and u ≤ v. Then, for all x, y ∈ X , x  y ⇐⇒ U (x, y) = u(x) − v(y) ≤ 0 ,

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which is null-diagonal and asymmetric but not skew-symmetric. Another representation is given by, for all x, y ∈ X , x  y ⇐⇒ UW (x, y) = (u(x) − v(y))+ − (u(y) − v(x))+ ≤ 0 , with W (x, y) = (u(x) − v(y))+ , which is skew-symmetric.

6.6 Representation Theorems with Utility Functions We may apply Debreu’s theorem in our restricted setting. Corollary 2 Assume (X, d) is a separable metric space. Let  ⊂ X 2 be a closed weak order. Then, there exists a continuous utility u : X −→ R, such that for all x, y ∈ X , x  y ⇐⇒ u(x) ≤ u(y) . Proof By Remark 2,  satisfies (CT). Since X is metric and separable it is second countable. Hence, by Debreu’s theorem there exists a continuous utility u representing .  In contrast, for non-separable metric spaces, a theorem of Estévez-Toranzo and Hervés-Beloso states the existence of continuous preferences which cannot be represented by a utility function ([17], see also [2] for further related results). We may now specialize the representation under a conservation condition in order to obtain a utility function representation. However, we know that a utility function representation may not always exist (see Example 2). W is conservative if for all x, y, z ∈ X , W (z, x) + W (x, y) + W (y, z) = W (z, y) + W (y, x) + W (x, z) . The conservation condition can be related to Chasles’ relation which states that one can consider “direct” cycle zx yz starting form z and passing first by x and then by y and then consider the “indirect” cycle zyx z starting form z and passing first by y and then by x without modifying the “overall flow”. We may notice that the conservation equation is symmetric in x, y, z. When W is asymmetric, at least three terms are null in the equation. Thus, one member contains at least two null terms. We may assume w.l.o.g. that W (x, y) = 0 and W (y, z) = 0. So, the conservation equation14 resumes to If W is nonnegative, asymmetric and conservative then it is null-transitive. Let x, y, z ∈ X with W (x, y) = W (y, z) = 0. If W (z, x) > 0 then W (x, z) = 0. Otherwise, if W (z, x) = 0 then by the conservation equation 0 = W (z, y) + W (y, x) + W (x, z). And since W ≥ 0, we have W (x, z) = 0.

14

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W (x, y) = 0 and W (y, z) = 0 ⇒ W (z, x) = W (z, y) + W (y, x) + W (x, z) . For instance, let  be represented by a (continuous) function u : X −→ R. Then Wu (x, y) = (u(x) − u(y))+ for all x, y ∈ X defines a (continuous) flow which is conservative. For all x, y, z ∈ X , we have Wu (x, z) − Wu (z, x) − (Wu (y, z) − Wu (z, y)) = (u(x) − u(z))+ − (u(z) − u(x))+ − ((u(y) − u(z))+ − (u(z) − u(y))+ ) = (u(x) − u(z)) − ((u(y) − u(z)) = u(x) − u(y) = Wu (x, y) − Wu (y, x) . Corollary 3 Assume (X, d) is a metric space. Let  ⊂ X 2 be a closed total binary relation which is represented by a skew-symmetric preutility function U . If (U )+ is conservative then, there exists a continuous utility u : X −→ R, such that for all x, y ∈ X , x  y ⇐⇒ u(x) ≤ u(y) , thus  is a weak order. Proof According to Corollary 1 there exists a continuous null-diagonal and asymmetric U such that for all x, y ∈ X , x  y ⇐⇒ U (x, y) ≤ 0 ⇐⇒ (U (x, y))+ ≤ 0 ⇐⇒ (U (x, y))+ − (U (y, x))+ ≤ 0. Let us put W = (U )+ and UW (x, y) = W (x, y) − W (y, x) for x, y ∈ X . Then, for all x, y ∈ X , x  y ⇐⇒ UW (x, y) ≤ 0. Assume that W = (U )+ is conservative. Fix some o ∈ X . Then, for all x, y ∈ X , W (o, x) + W (x, y) + W (y, o) = W (o, y) + W (y, x) + W (x, o) . Hence, for all x, y ∈ X , UW (x, y) = W (o, y) + W (x, o) − (W (o, x) + W (y, o)) = UW (x, o) − UW (y, o) . Hence, for all x, y ∈ X , x  y ⇐⇒ UW (x, y) ≤ 0 ⇐⇒ UW (x, o) ≤ UW (y, o).  So we may take u = UW (., o) to represent , which is continuous. Conservative flows and utility functions are in a 1-1 correspondence. We can identify conservative flows with utility functions. Fix some o ∈ X . Let W be a conservative flow. Then, for all x, y ∈ X we have WUW (.,o) (x, y) = (UW (x, o) − UW (y, o))+ = (UW (x, y))+ = W (x, y) . Yet we already know that for any utility function u we have that Wu (x, y) = (u(x) − u(y))+ for all x, y ∈ X defines a conservative flow and that UWu (x, y) = u(x) − u(y) for all x, y ∈ X . The conservation condition for flows can be related directly to Sincov’s functional equation for bivariate functions (see [16] for an historical perspective). Let

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F : X × X −→ R be defined. Then, F satisfies Sincov equation (see Proposition 1 in [6] and Proposition 6.10 p. 75 in [7]) if, for all x, y, z ∈ X, F(x, y) + F(y, z) = F(x, z) . Clearly, to any conservative flow W we may associate F(x, y) = UW (x, y) = W (x, y) − W (y, x) for x, y ∈ X which satisfies Sincov equation. Our Corollary 3 is an analog for metric spaces to Proposition 5 in [6] and Theorem 6.11 in [7] which relate total preorders that are representable by a utility function and Sincov equation. Theorem 5 (see Proposition 5 in [6], Theorem 6.11 p. 75 in [7]) Let X be a nonempty set. Let  be a total preorder on X . Then the following statements are equivalent: (i) The total preorder  is representable by means of a utility function u : X → R such that x  y ⇐⇒ u(x) ≤ u(y) (x, y ∈ X ). (ii) There exists a real-valued bivariate map F : X × X −→ R that satisfies the Sincov functional equation and, in addition, x ≺ y ⇐⇒ F(x, y) > 0 holds for every x, y ∈ X .

6.7 Microeconomics A natural question, for instance, is the following: How can preutility functions be useful in microeconomics? Moreover, can we give some interpretative content to preutilities? Firstly, in relation with optimization issues, one can retrieve standard utility maximization techniques as soon as we consider compact sets. Secondly, we can give a game theoretic content interpretation to preutility functions. Next proposition is connected to Bergstrom’s theorem for acyclic strict preferences [3]. Other finer results dealing with acyclic relations can be considered (see [24] and references therein). We shall deal with transitivity of the strict preference which is a sufficient condition in our setting. For instance, (closed) interval orders possess transitive strict preferences (see also [19]). Proposition 1 Assume (X, τ ) is a topological space. Let  ⊂ X 2 be a closed total binary relation. Let C = ∅ be a compact subset of X . Assume ≺ is transitive on C. Then, C admits a maximum (resp. minimum) xC∗ (x∗C ) ∈ C, i.e. x  xC∗ (resp. x∗C  x) for all x ∈ C. Proof We provide a proof for sake of completeness. Let us use the compact class theorem. Let C = ∅ be a compact subset of X . If for all finite subsets I = {x1 , . . . , xn } ⊂ C we have that max I  exists then K I = ∩i∈I [xi , →) ∩ C = ∅. Since  satisfies (USC), K I is compact. Hence, ∩x∈C [x, →) ∩ C = ∩ I K I = ∅. It remains to check that K I = ∅ for all finite subset I = {x1 , . . . , xn } ⊂ C. Since  is reflexive, any singleton {x} admits for maximum x. Let us proceed by induction. Let x1 , . . . , xn , xn+1 ∈ C with n ≥ 1. By induction hypothesis {x1 , . . . , xn } admits

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a maximum, say x n . If xn+1  x n then x n is a maximum in {x1 , . . . , xn , xn+1 }. Otherwise, we have xn+1 xr n by totality. If there exists some i ∈ {1, . . . , n} such that xi  xn+1 , then x n ≺ xn+1 and xn+1 ≺ xi . Thus, by transitivity of ≺ we have x n ≺ xi , a contradiction with x n being a maximum. Hence, for all i = 1, . . . , n, we have xi  xn+1 , i.e. xn+1 is a maximum in {x1 , . . . , xn , xn+1 }. The proof is similar for minimality.  Let us provide a possible interpretation of preutility functions through game theory. To be precise, we may identify preutility functions as a subclass of two players symmetric zero-sum games [20]. Let us recall that a two players zero-sum game consists of sets of strategies X and Y for player 1 and 2 respectively and an outcome function g : X × Y −→ R where g(x, y) is the gain of player 1 and −g(x, y) is the gain of player 2. To be slightly more general, one can consider antagonistic games where the players outcome functions have opposite monotonicity, i.e. ∀x, x  ∈ X , ∀y, y  ∈ Y , g1 (x, y) ≤ g1 (x  , y  ) ⇐⇒ g2 (x, y) ≥ g2 (x  , y  ) . A couple (x, ˜ y˜ ) is an equilibrium of the game G (X, Y, g) if α = g(x, ˜ y˜ ) = max min g(x, y) = min max g(x, y) x∈X y∈Y

y∈Y x∈X

where α ∈ R denotes the value15, .16 We have the following well-known property, Property 3 Let C = ∅ be a subset of X and U : X × X −→ R be a skew-symmetric function. If the two players zero-sum game G (C, C, U ) admits an equilibrium then its value is zero, i.e. α = 0. Proof Assume the game G (C, C, U ) has value α. Then, α = max min U (x, y) = min max U (x, y) . x∈C y∈C

y∈C x∈C

And also, α = min max U (x, y) = min max −U (y, x) = min(− min U (y, x)) y∈C x∈C

y∈C x∈C

y∈C

x∈C

= − max min U (y, x) = − max min U (x, y) = −α . y∈C x∈C

x∈C y∈C

As soon as X, Y are compact sets and g is a continuous function of (x, y), the minima, maxima, minimaxima, maximinima are all well defined. 16 Equilibrium is an ordinal property, i.e. if g  = φ ◦ g with φ increasing and ( x, ˜ y˜ ) is an equilibrium then (x, ˜ y˜ ) remains an equilibrium of the game G (X, Y, g  ) with value φ(α). 15

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So, 2α = 0 obtains, and this gives α = 0.



Finally, we may relate equilibria with extrema, Proposition 2 Assume (X, d) is a metric space. Let  ⊂ X 2 be a closed total binary relation represented by a continuous skew-symmetric preutility function U . Let C = ∅, C ⊂ X . Then, if  admits xC∗ ∈ C as a maximum in C then (xC∗ , xC∗ ) is an equilibrium of the game G (C, C, U ). Conversely, if (x˜C , y˜C ) is an equilibrium of the game G (C, C, U ) then x˜C and y˜C are maxima in C for . In particular, if C is a compact subset of X and U satisfies U (x, y) < 0, U (y, z) < 0 ⇒ U (x, z) < 0 for all x, y, z ∈ C, then the maxima and equilibria both exist. Proof Assume  admits xC∗ ∈ C as a maximum in C. Then, for all x ∈ C, we have U (x, xC∗ ) ≤ 0 and U (xC∗ , xC∗ ) = 0, thus maxx∈C U (x, xC∗ ) = 0. So, inf y∈C maxx∈C U (x, y) ≤ 0. Now for any y ∈ C, we have maxx∈C U (x, y) ≥ U (y, y) = 0. So, inf y∈C maxx∈C U (x, y) = 0, and in particular for y = xC∗ we obtain maxx∈C U (x, xC∗ ) = U (xC∗ , xC∗ ) = 0. So, inf y∈C maxx∈C U (x, y) = min y∈C maxx∈C U (x, y) = 0. Similarly, for all y ∈ C, we have U (xC∗ , y) ≥ 0 and U (xC∗ , xC∗ ) = 0, thus min y∈C U (xC∗ , y) = 0. So, supx∈C min y∈C U (x, y) ≥ 0. Now for any x ∈ C, we have min y∈C U (x, y) ≤ U (x, x) = 0. So, supx∈C min y∈C U (x, y) = 0, and in particular for x = xC∗ we obtain min y∈C U (xC∗ , y) = U (xC∗ , xC∗ ) = 0. So, supx∈C min y∈C U (x, y) = maxx∈C min y∈C U (x, y) = 0. So the minimax and maximin coincide, and then (xC∗ , xC∗ ) is an equilibrium. (Converse). Let (x˜C , y˜C ) be an equilibrium of the game G (C, C, U ). Then, according to Property 3, we have 0 = U (x˜C , y˜C ) = max min U (x, y) = min U (x˜C , y) , x∈C y∈C

y∈C

so U (x˜C , y) ≥ 0 for all y ∈ C, that is U (y, x˜C ) ≤ 0 for all y ∈ C, thus x˜C is a maximum in C for . And also, we have 0 = U (x˜C , y˜C ) = min max U (x, y) = max U (x, y˜C ) , y∈C x∈C

x∈C

so U (x, y˜C ) ≤ 0 for all x ∈ C, thus y˜C is a maximum in C for . (In particular). The condition U (x, y) < 0, U (y, z) < 0 ⇒ U (x, z) < 0 for all x, y, z ∈ C is equivalent to transitivity of ≺. According to Proposition 1, if C is compact and  is a closed total binary relation with ≺ transitive on C then  admits maxima and thus equilibria too.  We may notice that the maximum xC∗ guarantees to both players 1 and 2 nonnegative rewards. So, xC∗ is an optimal strategy for both players. We may extend the previous result to non-symmetric zero-sum games.

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Proposition 3 Assume X, Y are nonempty compact subsets of some metric spaces. Let g : X × Y −→ R be a continuous outcome function. Consider the binary relation U g represented by U g defined on X × Y as follows, U g ((x 1 , y 1 ), (x 2 , y 2 )) = g(x 1 , y 2 ) − g(x 2 , y 1 ), for all x 1 , x 2 ∈ X, y 1 , y 2 ∈ Y . Then, U g admits (x ∗ , y ∗ ) for maximum in X × Y if and only if (x ∗ , y ∗ ) is an equilibrium of the game G (X, Y, g) with value α = g(x ∗ , y ∗ ). In particular, if g satisfies the strict hexagon condition, i.e. ∀x 1 , x 2 , x 3 ∈ X, ∀y 1 , y 2 , y3 ∈ Y , g(x 1 , y 2 ) < g(x 2 , y 1 ) and g(x 2 , y 3 ) < g(x 3 , y 2 ) ⇒ g(x 1 , y 3 ) < g(x 3 , y 1 ) , then the maxima and equilibria both exist. Proof Let us consider the symmetric game G (X × Y, X × Y, U g ) where U g ((x 1 , y 1 ), (x 2 , y 2 )) = g(x 1 , y 2 ) − g(x 2 , y 1 ), for all x 1 , x 2 ∈ X, y 1 , y 2 ∈ Y . By construction, U g is skew-symmetric and continuous. Hence we may apply Proposition 2. Assume U g admits (x ∗ , y ∗ ) as a maximum in X × Y . Then, for all (x, y) ∈ X × Y , we have U g ((x, y), (x ∗ , y ∗ )) ≤ 0 which is equivalent to g(x, y ∗ ) − g(x ∗ , y) ≤ 0. That is, (x ∗ , y ∗ ) is an equilibrium. Conversely, assume (x ∗ , y ∗ ) is an equilibrium. Then, for all (x, y) ∈ X × Y , we have g(x, y ∗ ) − g(x ∗ , y) ≤ 0 which is equivalent to U g ((x, y), (x ∗ , y ∗ )) ≤ 0. That is, (x ∗ , y ∗ ) is a maximum in X × Y . (In particular). Let us check that transit-strict-negativity of U g is equivalent to the strict hexagon condition. Indeed, for x 1 , x 2 , x 3 ∈ X, y 1 , y 2 , y 3 ∈ Y , we have U g ((x 1 , y 1 ), (x 2 , y 2 )) < 0 ⇐⇒ g(x 1 , y 2 ) < g(x 2 , y 1 ), and U g ((x 2 , y 2 ), (x 3 , y 3 )) < 0 ⇐⇒ g(x 2 , y 3 ) < g(x 3 , y 2 ) . Then, by transit-strict-negativity or by the strict hexagon condition we obtain respectively, U g ((x 1 , y 1 ), (x 3 , y 3 )) < 0 or g(x 1 , y 3 ) < g(x 3 , y 1 ) . Hence according to Proposition 2, the maxima and equilibria both exist.



In comparison with classical minimax theorems, neither (quasi) convexity or concavity of the partial functions g(., y) and g(x, .) nor convexity of X, Y are required. However, this can be made possible whenever g satisfies the strict hexagon condition, this restricts severely the subclass of two players zero-sum games under scope.

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6.8 Conclusion We have elaborated a simple representation theorem for closed reflexive binary relations on metric spaces. The use of mathematics is reduced to standard metric concepts such as the point-to-set functions. Other topological notions such as connectedness, separability or second countability are not required anymore. Some by-products for weak orders representable by a continuous utility function are then deduced from a conservation condition and shown to be related to Sincov’s Equation. An interpretation of preutility functions is given through two players zero-sum symmetric games. A specification of these results on metric spaces to particular preferences dealing with intransitivities (e.g. semi-orders) could be envision in this relaxed setting through appropriate bivariate functions. Another line of research would be to consider topological versions for preferences representable by preutility functions.

References 1. Beardon, A.F.: Representation of continuous preferences. Econ. Theory 10(2), 369–372 (1997) 2. Beardon, A.F., Candeal, J.C., Herden, G., Induraín, E., Mehta, G.B.: The non-existence of a utility function and the structure of non-representable preference relations. J. Math. Econ. 37, 17–38 (2002) 3. Bergstrom, T.C.: Maximal elements of acyclic relations on compact spaces. J. Econ. Theory 10, 403–404 (1975) 4. Bridges, D., Mehta, G.B.: Representations of Preferences Orderings. Springer, Berlin, Heidelberg (1995) 5. Campión, M.-J., Gómez-Polo, C., Induraín, E., Raventós-Pujol, A.: A survey on the mathematical foundations of axiomatic entropy: representability and orderings. Axioms 7(29), 1–37 (2018) 6. Campión, M.-J., De Miguel, L., Catalán, R.G., Induraín, E., Abrisquetá, F.J.: Binary relations coming from solutions of functional equations: orderings and fuzzy subsets. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 25, 19–42 (2017) 7. Campión, M.-J., Induraín, E.: Open questions in utility theory. In: Bosi, G. et al. (eds.) Mathematical Topics on Representations of Ordered Structures and Utility Theory, vol. 263, pp. 47–81. Studies in Systems, Decision and Control. Springer (2020) 8. Chateauneuf, A.: Continuous representation of a preference relation on a connected topological space. J. Math. Econ. 16, 139–146 (1987) 9. Debreu, G.: Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Wiley, New York (1959) 10. Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R., Coombs, C., Davis, R. (eds.) Decision processes, pp. 159–165. Wiley, New York (1954) 11. Debreu, G.: Continuity properties of paretian utility. Int. Econ. Rev. 5, 285–293 (1964) 12. Debreu, G.: Economic theory in the mathematical mode. Scand. J. Econ. 86, 393–410 (1984) 13. Dugundji, J.: Topology. Allyn and Bacon, Boston (1966) 14. Eilenberg, S.: Ordered topological spaces. Am. J. Math. 63(1), 39–45 (1941) 15. Fishburn, P.C.: Interval representations for interval orders and semiorders. J. Math. Psychol. 10, 91–105 (1973) 16. Gronau, D.: A remark on Sincov’s functional equation. Not. South Afr. Math. Soc. 31(1), 1–8 (2000)

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17. Estevéz-Toranzo, M., Hervés-Beloso, C.: On the existence of continuous preference orderings without utility representations. J. Math. Econ. 24(4), 305–309 (1995) 18. Hervés-Beloso, C., del Valle-Inclán, Cruces H.: Continuous preference orderings representable by utility functions. J. Econ. Surv. 33, 1–17 (2019) 19. Kukushkin, N.: Maximizing an interval order on compact subsets of its domain. Math. Soc. Sci. 56, 195–206 (2008) 20. Rapoport, A.: Two-Person Game Theory. Dover, New York (1999) 21. Rébillé, Y.: Continuous utility on connected separable topological spaces. Econ. Theory Bull. 7(1), 147–153 (2019) 22. Rébillé, Y.: Representations of interval orders on connected separable topological spaces. In: Bosi G. et al. (eds.) Mathematical Topics on Representations of Ordered Structures and Utility Theory, vol. 263, pp. 85–108. Studies in Systems, Decision and Control. Springer (2020) 23. Shafer, W.J.: The nontransitive consumer. Econom 42(5), 913–919 (1974) 24. Zuanon, M.: A note on maximal elements for acyclic binary relations on compact topological spaces. Int. Math. Forum. 4, 537–541 (2009)

Chapter 7

Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems Santanu Acharjee, Kabindra Goswami, and Hemanta Kumar Sarmah

Abstract Topological entropy is a measure of complexity in topological dynamical systems. Recently, Acharjee et al. [S. Acharjee, K. Goswami and H.K. Sarmah, Transitive maps in bitopological dynamical systems, Filomat. 35(6), 2011–21(2021)] introduced bitopological dynamical systems to study dynamical systems with respect to two topologies. Also, Acharjee et al. [S. Acharjee, K. Goswami and H.K. Sarmah, On entropy of pairwise continuous map in bitopological dynamical systems, Commun. Math. Biol. Neurosci. 2020(2020), Article ID 81.] introduced the notion of entropy in bitopological dynamical systems where the underlying bitopological space was considered as weakly pairwise compact. In this chapter, we introduce entropy in non-weakly pairwise compact bitopological dynamical systems (in short NWPC bitopological dynamical systems) as a measure of complexity and produce several new results related to entropy. Also, we discuss about the possible connection between the neural activity of the human brain and the entropy of a pairwise continuous map in NWPC bitopological dynamical systems. Keywords Bitopological dynamical systems · Pairwise continuous map · Entropy · Neural activity

7.1 Introduction Entropy is a widely used measure of complexity and information content. The term ‘Entropy’ was used in many different fields viz. thermodynamics [1], inforS. Acharjee (B) · H. K. Sarmah Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India e-mail: [email protected] H. K. Sarmah e-mail: [email protected] K. Goswami Department of Mathematics, Goalpara College, Goalpara 783101, Assam, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_7

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mation theory [2] with different meanings before it was used in dynamical systems. Kolmogorov [3, 4] introduced the concept of entropy in dynamical systems. Later on, Sinai [5] introduced the Kolmogorov-Sinai entropy, a general version of entropy introduced by Kolmogorov. Adler et al. [6] introduced the concept of topological entropy in a compact topological space as an invariant for continuous mappings. In general, topological entropy is used to determine the complexity of a dynamical system [7]. According to [8, 9], if the topological entropy of a system is positive, then the system is topologically chaotic. Topological entropy has been applied in many areas viz. knot theory [10], billiard flows [11], DNA sequence analysis [12], etc. After Adler et al. [6], many researchers extended the notion of topological entropy to non-compact topological spaces as well as to non-compact topological dynamical systems. In 1973, Bowen [13] introduced topological entropy for non-compact sets. Later, Hofer [14] introduced two definitions of topological entropy for non-compact sets. For other works of entropy on non-compact topological space, one may refer to Hofer [15], Fedeli [16], Cánovas [17], and many others. Bitopological space is a widely studied concept of mathematics. It was introduced by Kelly [18]. Bitopological space has been successfully applied in many areas viz. medical sciences [19], economics [20, 21], computer science [22], etc. For recent works in bitopological space, one may refer to Acharjee et al. [23], Acharjee and Tripathy [24], Acharjee et al. [25], and many others. Bitopological dynamical system is a new area in the theory of dynamical systems recently introduced by Acharjee et al. [26]. Due to the inherent two topologies, bitopological dynamical system allows the study of dynamical properties of two parallel states (represented by two topologies) at a time. Recently, bitopological dynamical system has been applied to study human embryo development [26]. Acharjee et al. [26] proved that the child birth process from the zygote until birth is a bitopological dynamical system and the forward orbit of the zygote is point transitive. For other works in bitopological dynamical systems, one may refer to [27, 28]. Acharjee et al. [29] introduced the notion of entropy in bitopological dynamical systems where the underlying bitopological space was considered as weakly pairwise compact. But till now no work has been done to study the notion of entropy in bitopological dynamical systems where the underlying bitopological space is nonweakly pairwise compact. This inspires us for this chapter. We were motivated by the following fact: the real line R with the usual topology is not a compact topological space but its subsets of the form [a, b], where a and b are real numbers, is a compact topological space. This chapter is divided into mainly four sections. In the preliminary section, we recall some existing definitions of entropy, bitopological space and bitopological dynamical systems. In the next section, we introduce the notion of entropy of a pairwise continuous map in non-weakly pairwise compact bitopological dynamical systems. Then, we discuss some fundamental properties of entropy. We observe that most of the fundamental properties of entropy in bitopological dynamical systems,

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where the underlying bitopological space is weakly pairwise compact, is also true for our definition of entropy in non-weakly pairwise compact bitopological dynamical systems. Finally, we discuss the possible connection between the neural activity of the human brain and the entropy of a pairwise continuous map in NWPC bitopological dynamical systems.

7.2 Preliminary Definitions First, we recall some existing definitions which will be useful for us throughout this chapter. Definition 1 ([18]) A space X on which are defined two (arbitrary) topologies τ1 and τ2 is called a bitopological space and denoted by (X, τ1 , τ2 ). Definition 2 ([30]) A function f from a bitopological space (X, τ1 , τ2 ) into a bitopological space (Y, ψ1 , ψ2 ) is said to be pairwise continuous (respectively, a pairwise homeomorphism) if the induced functions f : (X, τ1 ) → (Y, ψ1 ) and f : (X, τ2 ) → (Y, ψ2 ) are continuous (respectively, homeomorphisms). Pervin [31] called this a continuous map. However, we call this as pairwise continuous map, due to Reilly [30]. Definition 3 ([32]) A set A ⊂ X is +invariant when f (A) ⊂ A and A is −invariant when A ⊂ f (A). A is called invariant when f (A) = A. Definition 4 ([27]) A map f : X → X is called +invariant if for all A ⊂ X , f (A) ⊂ A and −invariant when A ⊂ f (A). The map f is invariant when f (A) = A, for all A ⊂ X. In [26], we considered N, Z and R as the set of non-negative integers, the set of integers and the set of real numbers, respectively. Definition 5 ([26]) Let (X, τ1 , τ2 ) be a bitopological space. A bitopological dynamical system is a pair (X, f ), where (X, τ1 , τ2 ) is a bitopological space and f : X → X is a pairwise continuous map. The dynamics are obtained by iterating the map. The forward orbit of a point x ∈ X under f is defined as O+ (x) = { f n (x) : n ∈ N}, where f n denote the nth iteration of the map f . If f is a homeomorphism, then the backward orbit of x is the set O− (x) = { f −n (x) : n ∈ N} and the full orbit of x (or simply orbit of x) is the set O(x) = { f n (x) : n ∈ Z}. Here, homeomorphism of f indicates homeomorphism of the function f : (X, τi ) → (X, τi ) separately for all i ∈ {1, 2} as it is clear in terms of bitopological space. Thus, we can consider pairwise homeomorphism equivalently. For a compact topological space, Adler et al. [6] introduced the concept of topological entropy and studied its various properties. Their definition is as follows:

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Definition 6 ([6]) Let X be a compact topological space and f : X → X a continuous mapping. For any open cover U of X , let N (U ) denote the number of sets in a subcover of minimal cardinality. Let H (U ) = log N (U ). For any two open covers U and V ; U ∨ V ≡ {A ∩ B : A ∈ U , B ∈ V } defines their join. Then, the entropy ent ( f, U ) of a mapping f with respect to a cover U is defined as H (U ∨ f −1 U ∨ · · · ∨ f −n+1 U ) n→∞ n

ent ( f, U ) = lim

and the entropy ent ( f ) of a mapping f is defined as the sup ent ( f, U ), where the supremum is taken over all open covers U . For a non-compact topological space, Hofer [14] introduced two definitions of topological entropy. Definition 7 ([14]) Let (X, τ ) be a non-compact Hausdorff space, and let f : X → X be a homeomorphism. The entropy ent ∗ ( f ) of the mapping f is defined by the formula ent ∗ ( f ) = ent ( f ∗ ), where f ∗ is the unique continuous extension of f ˇ to the Stone-Cech compactification X ∗ of X . Here, (X, τ ) is a completely regular topological T1 −space and ent ( f ∗ ) denote the entropy of the map f ∗ as defined by Adler et al. [6]. Definition 8 ([14]) Let (X, τ ) be a non-compact Hausdorff space, and let f : X → X be a homeomorphism. The entropy ent ∗∗ ( f ) of the mapping f is defined by the formula ent ∗∗ ( f ) = sup{ent ( f, U )} U

where the supremum is taken over all finite open covers U of X . Here, ent ( f, U ) is same as defined in Definition 6. Recently, Acharjee et al. [29] introduced the notion of entropy in bitopological dynamical system where the underlying bitopological space is weakly pairwise compact. Their definition is as follows: Definition 9 ([29]) A cover U of a bitopological space (X, τ1 , τ2 ) is weakly pairwise open if U ⊂ τ1 ∪ τ2 ∪ {U ∩ V : U (= ∅, X ) ∈ τ1 , V (= ∅, X ) ∈ τ2 }, U ∩ τ1 contains a non-empty set and U ∩ τ2 contains a non-empty set. Definition 10 ([29]) A bitopological space (X, τ1 , τ2 ) is weakly pairwise compact provided every weakly pairwise open cover of X has a finite subcover. Definition 11 ([29]) Let (X, f ) be a bitopological dynamical system, where (X, τ1 , τ2 ) is a bitopological space and f : X → X is a pairwise continuous map. Let (X, τ1 , τ2 ) be weakly pairwise compact. For any weakly pairwise open cover U of X , let N ∗ (U ) denote the number of sets in a subcover of U with minimal cardinality. Let L ∗ (U ) = log N ∗ (U ). For any two weakly pairwise open covers U and V of X , we define their join by U ∨ V = {U ∩ V : U ∈ U , V ∈ V }.

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Definition 12 ([29]) Let (X, f ) be a bitopological dynamical system, where the bitopological space X is weakly pairwise compact. We define the entropy of the mapping f relative to the weakly pairwise open cover U as  n−1 1 ∗  −i E ( f, U ) = lim L f (U ) n→∞ n i=0 ∗

. Definition 13 ([29]) Let (X, f ) be a bitopological dynamical system, where the bitopological space X is weakly pairwise compact. The entropy of f is defined as E ∗ ( f ) = sup{E ∗ ( f, U )}, where the supremum is taken over all weakly pairwise U

open covers U of X . This entropy is called as bitopological entropy. Definition 14 Let (X, f ) be a bitopological dynamical system, where the bitopological space X is not weakly pairwise compact. We call (X, f ) as a non-weakly pairwise compact bitopological dynamical system (in short NWPC bitopological dynamical system). Lemma 1 ([8]) Let { pn } be a sequence of real numbers which is subadditive, i.e. pm+n ≤ pm + pn for all m, n. Then lim pnn exists and has the value c = inf pnn . n→∞

Theorem 1 ([6]) Let X and Y be two compact topological spaces. Let φ1 be a continuous mapping of X into itself and φ2 a continuous mapping of Y into itself. Then, ent (φ1 × φ2 ) = ent (φ1 ) + ent (φ2 ) where φ1 × φ2 is the continuous mapping of X × Y into itself defined by φ1 × φ2 : (x, y) → (φ1 x, φ2 y).

7.3 On Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems In this section, our main aim is to introduce entropy in NWPC bitopological dynamical systems. Let N, Z and R denote the set of non-negative integers, the set of integers and the set of real numbers, respectively. Definition 15 Let (X, f ) be a bitopological dynamical system, where (X, τ1 , τ2 ) is a bitopological space and f : X → X is a pairwise continuous map. Let (X, τ1 , τ2 ) be a non-weakly pairwise compact bitopological space. Let K (X ) denote the set of a countable number of non-empty weakly pairwise compact subsets of X such that each K ∈ K (X ) is either +invariant or invariant for the map f . For K ∈ K (X ) and

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for weakly pairwise open cover U of X , let N K (U ) denotes the number of sets in a subcover (for K ) of U with minimal cardinality. Also, let L K (U ) = log N K (U ). For any two weakly pairwise open covers U and V of X , we define their join by U ∨ V = {U ∩ V : U ∈ U , V ∈ V }. Then, U ∨ V is also a weakly pairwise open cover of X . A weakly pairwise open cover U is said to be a refinement of a weakly pairwise open cover V , denoted by V ≺ U , if every member of U is a subset of at least one member of V . Following are some basic properties of a weakly pairwise open cover. Property 1 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). For any weakly pairwise open cover U of X , N K (U ) ≥ 1, and therefore, L K (U ) ≥ 0. Proof There are two possibilities: (a) The subcover (for K ) of U with minimal cardinality contains K and (b) The subcover (for K ) of U with minimal cardinality does not contain K . In first case, we will have N K (U ) = 1 and in the second case N K (U ) > 1. So,  N K (U ) ≥ 1 and thus L K (U ) = log N K (U ) ≥ 0. Property 2 Let (X, f ) be a NWPC bitopological dynamical system, K ∈ K (X ) and U be any weakly pairwise open cover of X . Then, L K ( f −1 (U )) ≤ L K (U ). Proof Let U ∗ = {U1 , . . . , Ur , V1 , . . . , Vs , A1 ∩ B1 , . . . , A p ∩ Bq : U j , Ai ∈ τ1 ; Vl , Bk ∈ τ2 where i = 1, 2, . . . , p; j = 1, 2, . . . r ; k = 1, 2, . . . , q and l = 1, 2, . . . , s} be a subcover (for K ) of U with minimal cardinality. Then, due to pairwise continuity of f , f −1 (U ∗ ) = { f −1 (U ) : U ∈ U ∗ } is also a weakly pairwise open cover and clearly, it covers K . But, this subcover is possibly not the minimal subcover (for K ) of f −1 (U ). Hence, N K ( f −1 (U )) ≤ N K (U ). This gives L K ( f −1 (U )) = log N K ( f −1 (U )) ≤ log N K (U ) = L K (U ). Thus, L K ( f −1 (U )) ≤ L K (U ).



Property 3 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). For any weakly pairwise open cover U of X , L K (U ∨ V ) ≤ L K (U ) + L K (V ). Proof Let U ∗ = {A1 , . . . , A p } be a subcover (for K ) of U with minimal cardinality and V ∗ = {B1 , . . . , Bq } be a subcover (for K ) of V with minimal cardinality, i.e. N K (U ) = p and N K (V ) = q, where Ai ’s and B j ’s are either τ1 -open sets or τ2 open sets or sets of the form U ∩ V , U ∈ τ1 , V ∈ τ2 . Now, U ∗ ∨ V ∗ = {Ai ∩ B j : i = 1, . . . , p and j = 1, . . . , q} is a subcover (for K ) of U ∨ V which contains pq elements. But, this subcover is possibly not the minimal subcover (for K ) of U ∨ V . Thus, N K (U ∨ V ) ≤ pq. This implies

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L K (U ∨ V ) = log N K (U ∨ V ) ≤ log pq = log p + log q = log N K (U ) + log N K (V ) = L K (U ) + L K (V ). Hence, L K (U ∨ V ) ≤ L K (U ) + L K (V ).



Property 4 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). Then for any two weakly pairwise open covers U and V of X , f −1 (U ∨ V ) = f −1 (U ) ∨ f −1 (V ). Proof Let U = {Ai , i ∈ Δ, Δ is an index set} and V = {B j , j ∈ ,  is an index set} be two weakly pairwise open covers of X , where Ai ’s and B j ’s are either τ1 open sets or τ2 -open sets or sets of the form U ∩ V , U ∈ τ1 , V ∈ τ2 . Now, U ∨ V = {Ai ∩ B j : Ai ∈ U , B j ∈ V }. Using properties of the inverse of a function, we get f −1 (U ∨ V ) = { f −1 (Ai ∩ B j ) : Ai ∈ U , B j ∈ V } = { f −1 (Ai ) ∩ f −1 (B j ) : Ai ∈ U , B j ∈ V } = f −1 (U ) ∨ f −1 (V ). Thus, f −1 (U ∨ V ) = f −1 (U ) ∨ f −1 (V ).



Property 5 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). Let U and V be two weakly pairwise open covers of X such that V ≺ U . Then, N K (V ) ≤ N K (U ) and so L K (V ) ≤ L K (U ). Proof Let U ∗ = {A1 , . . . , A p } be a subcover (for K ) of U with minimal cardinality, that is N K (U ) = p, where Ai ’s are either τ1 -open sets or τ2 -open sets or sets of the form U ∩ V , U ∈ τ1 , V ∈ τ2 . Now, since V ≺ U , so each Ai ∈ U is a subset of at least one member, say Bi of V , where Bi ’s are either τ1 -open sets or τ2 -open sets or sets of the form U ∩ V , U ∈ τ1 , V ∈ τ2 . Clearly, V ∗ = {B1 , . . . , B p } also covers K . But, this subcover of V may not be the minimal subcover (for K ) of V . Hence, N K (V ) ≤ p. This implies N K (V ) ≤ N K (U ). This gives L K (V ) = log N K (V ) ≤ log N K (U ) = L K (U ). Thus, L K (V ) ≤ L K (U ).



Property 6 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). Then for any weakly pairwise open cover U of X , the limit

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1 lim L K n→∞ n



n−1 

f

−i

(U )

i=0

exists and it is finite. Proof Let pn = L K

n−1 

 f

−i

(U ) .

i=0

For any two positive integers m and n, we have pm+n = L K

m+n−1 

 f −i (U )

i=0

= LK

m−1 

 f

−i

(U ) ∨

i=0

= LK

m−1 

 f

−i

≤ LK

f

f

 −i

 −i

f

(U )

i=m



(U ) ∨

i=0

m−1 

m+n−1 

−m

 −i

f

(U )

i=0



(U ) + L K

n−1 

f

−m

n−1 

i=0

 f

−i

(U ) , by property 3

i=0

= am + an . Also, property 2 states that for any weakly pairwise open cover U of X , L K ( f −1 (U )) ≤ L K (U ). This implies that L K ( f −i (U )) ≤ L K (U ), for any positive integer i. Hence, pm+n ≤ L K

m−1  i=0

≤ LK

m−1  i=0

 f

−i

(U ) + L K 

f −i (U ) + L K

 f

−m

n−1 

n−1 

 f

−i

i=0

(U )



f −i (U )

i=0

= pm + pn . Thus, pm+n ≤ pm + pn for all positive integers m, n. Now, using the Lemma 1 we have the limit  n−1  1 −i lim L K f (U ) n→∞ n i=0 exists and has the value

7 Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems

1 c = inf L K n

n−1 

139

 f

−i

(U )

i=0



which is finite. This completes the proof.

Now, we define the entropy of a pairwise continuous map in NWPC bitopological dynamical systems. Definition 16 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). The entropy of f on K relative to the weakly pairwise open cover U is defined as  n−1  1 −i E( f, U , K ) = lim L K f (U ) . n→∞ n i=0 Property 7 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). For any weakly pairwise open cover U of X , E( f, U , K ) ≤ L K (U ). Proof We have 1 E( f, U , K ) = lim L K n→∞ n

n−1 

 f

−i

(U )

i=0

1 (L K (U ) + · · · + L K ( f −(n−1) (U ))), using Property 3 n→∞ n 1 ≤ lim (L K (U ) + · · · + L K (U )), using Property 2 n→∞ n ≤ L K (U ).

≤ lim

Hence, E( f, U , K ) ≤ L K (U ).



Definition 17 Let (X, f ) be a NWPC bitopological dynamical system and K ∈ K (X ). The entropy of f on K is defined as E( f, K ) = sup{E( f, U , K )}, where U

the supremum is taken over all weakly pairwise open covers U of X. Property 8 Let (X, f ) be a NWPC bitopological dynamical system and K 1 , K 2 ∈ K (X ) be such that K 1 ⊆ K 2 . Then, E( f, K 1 ) ≤ E( f, K 2 ). Proof Let U be any weakly pairwise open cover of X . We first prove that E( f, U , K 1 ) ≤ E( f, U , K 2 ). Let {M1 , . . . , Mr } be a subcover (for K 2 ) of n−1 −i n−1 −i (U ) with minimal cardinality, that is N K 2 (U ) = r , where i=0 f i=0 f Mi ’s are either τ1 -open sets or τ2 -open sets or sets of the form U ∩ V, U ∈ τ1 , V ∈ n−1 −i f (U ). τ2 . As K 1 ⊆ K 2 , {M1 , . . . , Mr } is also a subcover (for K 1 ) of i=0 n−1 −i But, this subcover (for K1 ) of i=0 f (U ).  may not be the minimal subcover n−1 −i n−1 −i (U ) ≤ r = N K 2 (U ) . This implies Hence, N K1 i=0 f i=0 f     n−1 −i n−1 −i (U ) ≤ L K 2 (U ) . Therefore, L K1 i=0 f i=0 f

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 n−1  1 −i E( f, U , K 1 ) = lim L K 1 f (U ) n→∞ n i=0  n−1  1 −i ≤ lim L K 2 f (U ) n→∞ n i=0 = E( f, U , K 2 ) This gives E( f, K 1 ) = sup{E( f, U , K 1 )} U

≤ sup{E( f, U , K 2 )} U

= E( f, K 2 ). Hence, E( f, K 1 ) ≤ E( f, K 2 ).



Definition 18 Let (X, f ) be a NWPC bitopological dynamical system. When K (X ) = φ, the entropy of f is defined as E( f ) = sup {E( f, K )}. When K (X ) = φ, we define E( f ) = 0.

K ∈K (X )

7.4 Fundamental Properties of Entropy in NWPC Bitopological Dynamical Systems In this section, we discuss some fundamental properties of entropy in NWPC bitopological dynamical systems. Theorem 2 Let (X, f ) be a NWPC bitopological dynamical system, and let f be a +invariant map. Then, the entropy of f is 0. Proof Let K ∈ K (X ) be arbitrary. As f is +invariant, f (A) ⊂ A for all A ⊂ X , that is A ⊂ f −1 (A). Then for any weakly pairwise open cover U of X , f −1 (U ) ≺ U . This gives U ∨ f −1 (U ) ≺ U . By Property 5, N K (U ∨ f −1 (U )) ≤ N K (U ). Also, U ≺ U ∨ f −1 (U ). This gives N K (U ) ≤ N K (U ∨ f −1 (U )). Thus, N K (U ) = N K (U ∨ f −1 (U )). Now, replacing U by f −1 (U ) in above inequalities we get f −1 (U ) ∨ −1 f ( f −1 (U )) ≺ f −1 (U ); that is f −1 (U ) ∨ f −2 (U ) ≺ f −1 (U ). This implies U ∨ f −1 (U ) ∨ f −2 (U ) ≺ U ∨ f −1 (U ). By Property 5, N K (U ∨ f −1 (U ) ∨ f −2 (U )) ≤ N K (U ∨ f −1 (U )). Similarly, from the second inequality we get N K (U ∨ f −1 (U )) ≤ N K (U ∨ f −1 (U ) ∨ f −2 (U )). Hence, N K (U ) = N K (U ∨ f −1 (U )) = N K (U ∨ f −1 (U ) ∨ f −2 (U )). Continuing the  same procedure, for n−1 −i (U ) . This gives L K (U ) = any positive integer n we get N K (U ) = N K i=0 f

7 Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems

LK

 n−1 i=0

141

 f −i (U ) . Hence, 1 E( f, U , K ) = lim L K n→∞ n = lim

1

n→∞ n

n−1 

 f

−i

(U )

i=0

L K (U )

= 0. Thus, E( f, K ) = sup{E( f, U , K )} = 0 and so E( f ) = U

sup {E( f, K )} = 0. 

K ∈K (X )

Theorem 3 Let (X, f ) be a NWPC bitopological dynamical system and let f be the identity map. Then E( f ) = 0. We omit the proof of Theorem 3 as it can be proved similarly as property 3.8. of [29]. Theorem 4 Let (X, f ) be a NWPC bitopological dynamical system. For any positive integer k, E( f k ) = k E( f ). Proof If K (X ) = φ, the result is clearly true. We consider K (X ) = φ. Let U be −j any weakly pairwise open cover of X . Let V = k−1 (U ). Now, for K ∈ K (X ) j=0 f we have E( f k , K ) ≥ E( f k , V , K )  n−1  1 = lim L K ( f k )−i (V ) n→∞ n i=0 ⎞⎞ ⎛ ⎛ n−1 k−1   1 = lim L K ⎝ ( f k )−i ⎝ f − j (U )⎠⎠ n→∞ n i=0 j=0  n−1 

1 k −i −1 −k+1 = lim L K ( f ) U ∨ f (U ) ∨ · · · ∨ f (U ) n→∞ n i=0 1 L K (U ∨ f −1 (U ) ∨ · · · ∨ f −k+1 (U ) ∨ f −k (U )∨ n→∞ n f −k−1 (U ) ∨ · · · ∨ f −2k+1 (U ) ∨ · · · ∨ f −kn+k (U ) ∨ · · · ∨ f −kn+1 (U ))  kn−1  1 = lim L K f −i (U ) n→∞ n i=0  kn−1  1 −i LK = k lim f (U ) kn→∞ kn i=0 = lim

= k E( f, U , K ), for any weakly pairwise open cover U .

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Hence, E( f k , K ) ≥ k E( f, K ). Taking supremum over all K ∈ K (X ), we get E( f k ) ≥ k E( f ). Again, for any weakly pairwise open cover U of X , U ∨ ( f k )−1 (U ) ∨ · · · ∨ ( f k )−(n−1) (U ) ≺ U ∨ f −1 (U ) ∨ · · · ∨ f −nk+1 (U ) Hence, for K ∈ K (X ) and any arbitrary positive integer k E( f, K ) ≥ E( f, U , K )  n−1  1 f −i (U ) = lim L K n→∞ n i=0  km−1  1 −i LK = lim f (U ) , let n = km km→∞ km i=0  m−1  1 k −i LK ( f ) (U ) ≥ lim km→∞ km i=0  m−1  1 1 k −i = lim L K ( f ) (U ) k m→∞ m i=0 =

1 E( f k , U , K ), for any weakly pairwise open cover U . k

Thus, E( f, K ) ≥ k1 E( f k , K ). Taking supremum over all K ∈ K (X ); we get  E( f ) ≥ k1 E( f k ), that is k E( f ) ≥ E( f k ). Hence, E( f k ) = k E( f ).

7.5 Possible Connection to Neural Activity of Human Brain Acharjee et al. [26–28] applied bitopological dynamical systems to study human embryo development from zygote until birth and disproved three conjectures of Nada and Zohny [33]. In [26], authors found that the forward orbit of the zygote is point transitive. In another paper, Acharjee et al. [27] showed that the growth process of an organism from the zygote, after gastrulation, is both pairwise Hausdorff and forward iterated Hausdorff and they predicted theoretically that after the gastrulation; no structural relationship can be found between the growth of the brain together with the central nervous system and the other body parts except the brain and the central nervous system. In [28], Acharjee et al. showed that during the development of an organism from zygote until birth, the developing stage after gastrulation is pairwise disconnected and forward iterated disconnected. Also, authors found [28] that to cure cognitive anomalies medical treatments should be started on the brain cells or on the central nervous system cells disproving the conjecture 2 of Nada and Zohny [33]. Further, Acharjee et al. [29] calculated the bitopological entropy of the mitosis map

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in the bitopological space of the postgastrulation part of human embryo development and also they found the weighted bitopological Shannon entropy of this space. Now, in many aspects of human embryo development such as the development of neuron cells, we found the non-pairwise compact [29] nature of the corresponding bitopological space. According to [34, 35], after reaching of a signal to the brain for performing a certain task, only a few neurons become activate to perform the related neural activity. On the other hand, most of the neurons remain inactive to the specific task. Thus, the neural behavior of the human brain instigates us to think that the neural activity may not be related to weakly pairwise compactness [29] or pairwise compactness [36]. Thus, we suspect that neural behavior of the human brain as stated above may be somehow related to non-weakly pairwise compactness, thus to NWPC bitopological dynamical systems. So, a possible connection between the neural activity of the brain and the entropy of a pairwise continuous map in NWPC bitopological dynamical systems may be found in the future.

7.6 Open Questions In this section, we formulate some questions which are still open. More sophisticated theoretical results in bitopological dynamical system will be needed to handle these. The questions are given as follows: Q.1. Let (X, f ) and (Y, g) be two NWPC bitopological dynamical systems. Is the following result hold? E( f × g) = E( f ) + E(g) where f × g is the pairwise continuous mapping of X × Y into itself defined by f × g : (x, y) → ( f (x), g(y)). Q.2. Topological entropy acts as an invariant for continuous mappings. Does our notion of entropy in NWPC bitopological dynamical systems also satisfy similar property like ‘invariant’? We are expecting that answers to these questions will enrich our notion of entropy in NWPC bitopological dynamical systems.

7.7 Conclusion In this chapter, we introduced the notion of entropy of a pairwise continuous map in NWPC bitopological dynamical systems and discuss some fundamental properties of entropy in NWPC bitopological dynamical systems. We observe that most of the fundamental properties of entropy of a pairwise continuous map in weakly pairwise compact bitopological space [29] also hold for our definition of entropy in NWPC bitopological dynamical systems. So, from a theoretical point of view, we can pre-

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dict that our definition of entropy in NWPC bitopological dynamical systems is a good measure of complexity. We hope that our predictions will be found true in the future when NWPC bitopological dynamical systems will be studied by researchers thoroughly.

References 1. Clausius, R.: The Mechanical Theory of Heat. Macmillan (1879) 2. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948) 3. Kolmogorov, A.N.: A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR 119(5), 861–864 (1958) 4. Kolmogorov, A.N.: Entropy per unit time as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR 124, 754–755 (1959) 5. Sinai, Y.G.: On the notion of entropy of a dynamical system. Dokl. Akad. Nauk. SSSR 124, 768–771 (1959) 6. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114(2), 309–319 (1965) 7. Sang, W.K., Salleh, Z.: A note on the notions of topological entropy. Earthline J. Math. Sci. 1(1), 1–16 (2019) 8. Block, L.S., Coppel, W.A.: Dynamics in One Dimension. In: Lecture Notes in Mathematics. Springer, Berlin (1992) 9. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edn. CRC Press, Florida (1999) 10. Ghrist, R.W.: Chaotic knots and wild dynamics. Chaos, Solitons Fractals 9, 583–598 (1998) 11. Foltin, C.: Billiards with positive topological entropy. Nonlinearity 15, 2053–2076 (2002) 12. Koslicki, D.: Topological entropy of DNA sequences. Bioinformatics 27(8), 1061–1067 (2011) 13. Bowen, R.: Topological entropy for noncompact sets. Trans. Am. Math. Soc. 184, 125–136 (1973) 14. Hofer, J.E.: Topological entropy for noncompact spaces. Michigan Math. J. 21, 235–242 (1974) 15. Hofer, J.E.: Topological entropy for noncompact spaces and other extensions. Technical Report No. 45, Oregon State University, Department of Mathematics, Corvallis (1971) 16. Fedeli, A.: On two notions of topological entropy for noncompact spaces. Chaos, Solitons Fractals 40, 432–435 (2009) 17. Cánovas, J.S.: Entropy on noncompact sets. Topol. Algebr. Appl. 7, 29–37 (2019) 18. Kelly, J.C.: Bitopological spaces. Proc. Lond. Math. Soc. 13(3), 71–89 (1963) 19. Salama, A.S.: Bitopological rough approximations with medical applications. J. King Saud Univ. Sci. 22, 177–183 (2010) 20. Acharjee, S., Tripathy, B.C.: Strategies in mixed budget: a bitopological approach. New Math. Nat. Comput. 15(01), 85–94 (2019) 21. Bosi, G., Mehta, G.B.: Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. J. Math. Econ. 38, 311–328 (2002) 22. Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological duality for distributive lattices and Heyting algebras. Math. Struct. Comput. Sci. 20(3), 359–393 (2010) 23. Acharjee, S., Reilly, I.L., Sarma, D.J.: On compactness via bI -open sets in ideal bitopological spaces. Bull. Transilvania Univ. Brasov. Math, Inform. Phys. Ser. III 12, 1–8 (2019) 24. Acharjee, S., Tripathy, B.C.: p-J -generator and p1 -J -generator in bitopology. Bol. Soc. Paran. Math. 36(2), 17–31 (2018) 25. Acharjee, S., Papadopoulos, K., Tripathy, B.C.: Note on p1 -Lindelöf spaces which are not contra second countable spaces in bitopology. Bol. Soc. Paran. Math. 38(1), 165–171 (2020)

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26. Acharjee, S., Goswami, K., Sarmah, H.K.: Transitive maps in bitopological dynamical systems. Filomat 35(6), 2011–21 (2021) 27. Acharjee, S., Goswami, K., Sarmah, H.K.: On forward iterated Hausdorffness and development of embryo from zygote in bitopological dynamical systems. Bull. Transilvania Univ. BrasovMath., Inform., Phys. Ser. III. 13(62), 399–410 (2020) 28. Acharjee, S., Goswami, K., Sarmah, H.K.: On conjecture 2 of Nada and Zohny from the perspective of bitopological dynamical systems. J. Math. Comput. Sci. 11, 278–291 (2021) 29. Acharjee, S., Goswami, K., Sarmah, H.K.: On entropy of pairwise continuous map in bitopological dynamical systems. Commun. Math. Biol. Neurosci. 2020, Article ID 81 (2020) 30. Reilly, I.L.: On pairwise connected bitopological spaces. Kyungpook Math. J. 11(1), 25–28 (1971) 31. Pervin, W.J.: Connectedness in bitopological spaces. Indag. Math. 29, 369–372 (1967) 32. Akin, E., Carlson, J.D.: Conceptions of topological transitivity. Topol. Appl. 159, 2815–2830 (2012) 33. Nada, S.I., Zohny, H.: An application of relative topology in biology. Chaos, Solitons Fractals 42, 202–204 (2009) 34. Eagleman, D.: The Brain: The Story of You. Pantheon Books. First American Edition, New York (2015) 35. Sadler, T.W.: Langman’s Medical Embryology, 14th edn. Wolters Kluwer, Philadelphia (2019) 36. Fletcher, P., Hoyle, H.B., III., Patty, C.W.: The comparison of topologies. Duke Math. J. 36, 325–331 (1969)

Chapter 8

Topological Approaches for Vector Variational Inequality Problems Ankit Gupta, Satish Kumar, and Pankaj Kumar Garg

Abstract In this chapter, we discuss several variants of variational inequality problems that exist in the literature. We also discuss solutions to the vector variational inequality problem and generalized vector variational inequality problem in a more general framework, using a topological approach. We consider X and Y as topological vector spaces and provide different sets of conditions for the existence of solutions in the light of upper semi-continuity and lower semi-continuity, respectively. Admissibility of function space topology and convergence of net of sets are used as major tools towards achieving this goal. Topological properties of the solution sets of VVI and GVVI problems are also discussed. Keyword Variational inequality problems · Admissibility · Upper-semi continuity · Compactness · Continuous-convergence

8.1 Introduction In the early sixties, the theory of variational inequality came into the picture when Stampacchia [20] tackled the variational inequality problem while working on the study of the capacity of sets in the arena of potential theory. In the same period, Fischera’s work on variational inequality was motivated by the obstacle problems in elasticity also known as the Signorini problem. After the fundamental work of G. Stampacchia, the study of variational inequalities intensified and became an important part of research in the field of optimization, economics and non-linear analysis.

A. Gupta (B) Department of Mathematics, Bharati College (University of Delhi), Delhi 110058, India e-mail: [email protected] S. Kumar Department of Mathematics, University of Delhi, Delhi 110007, India P. K. Garg Department of Mathematics, Rajdhani College (University of Delhi), Delhi 110015, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_8

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In the eighties, a breakthrough in the finite dimensional theory came, when Dafermos [11] studied the traffic network equilibrium conditions as introduced by Smith [38] in the light of variational inequalities. He found that the traffic equilibrium conditions have a structure of variational inequalities. The finite dimensional vector variational inequality problems were first studied by Giannessi [15], where he extended the classical variational inequality problem for vector-valued functions. After this pioneering work, the interest in the investigation of the interrelationship between variational inequality and vector optimization problems has increased manifold. As a result, several conditions were explored for the existence of solutions for many variational-like inequalities having various applications in various fields of operational research, economics and control theory. Later, it was proved that the vector variational inequalities can be used to study vector optimization problems as well. Since then, vector variational inequality has been extended and generalized in many directions under various setups. The theory of vector variational inequalities was initiated by Giannessi [15] in 1980. In the last three decades, vector variational inequality problems and their generalization have been used as a tool to solve vector optimization problems [16, 21, 31, 32, 35, 41]. In this chapter, we try to pen down some generalizations of vector variational inequality problems under different frameworks and the existence of their solution sets. In the last section of this chapter, we use several topological concepts such as closedness and compactness to show the existence of solutions to vector variational inequality problems. We also discuss how net-theory and admissibility of function space topologies play a vital role in the existence of solutions to vector variational inequality problems and their generalizations.

8.2 Variational Inequalities and Their Generalizations In 1964, Hartman and Stampacchia [20] introduced the notion of the variational inequality problem in their seminal paper. Because of the utility of the topic in fundamental sciences, it became very popular amongst researchers. Variational Inequality Problem [20] Let K be a non-empty subset of Rn , and let f : K ⊆ Rn → Rn be a continuous function. Then the variational inequality problem is to find x0 ∈ K such that  f (x0 ), x − x0  ≥ 0 for all x ∈ K . The concept of vector variational inequality was first investigated and introduced by Giannessi for finite dimensional spaces. Here, he replaced the map f with a matrix-valued function. Vector Variational Inequality Problem [15] Let K be a non-empty convex subset of Rn , and let f : K ⊆ Rn → Rm×n be a vector-valued function. Then the vector variational inequality problem is to find x0 ∈ K such that there exists no

8 Topological Approaches for Vector Variational Inequality Problems

x ∈ K , satisfying

149

f (x0 )(x − x0 ) ≤ 0.

After the fruitful investigation of the vector variational inequality problem, Yang [40] and Siddiqi et al. [37] generalized the notion of vector variational inequality problem and investigated the same for various applications in vector optimization problems. They used the class of η-connected sets, which is more general than the class of convex sets. This generalized form of vector variational inequality problem was named the vector variational-like inequality problem. Vector Variational-Like Inequality Problem [37, 40] Let K be a non-empty subset of Rn ; η : K × K → Rn be a vector-valued function. Let f : K ⊆ Rn → Rm×n be a matrix-valued function. Then the vector variational-like inequality problem is to find x0 ∈ K such that there exists no x ∈ K , satisfying f (x0 )η(x, x0 ) ≤ 0. After this development, the vector variational inequality problem was further extended for Banach spaces, Normed spaces and topological vector spaces as well. Vector Variational Inequality Problem for Banach Spaces [10] Let X and Y be real Banach spaces and K be a non-empty closed and convex subset of X . Let C be a pointed, closed and convex cone in Y with non-empty interior, that is, intC = ∅. Let L (X, Y ) be the space of all continuous linear mapping from X to Y , and let f : K → L (X, Y ) be a single-valued function. Then the vector variational inequality problem over a Banach space is to find x0 ∈ K such that f (x0 )(x − x0 ) ∈ C for all x ∈ K . Later on, in the nineties, Chen and Craven [8] extended the idea of variational inequality problem for set-valued mappings, and this new notion was introduced as a generalized vector variational inequality problem. Generalized Vector Variational Inequality Problem [8] Let X be a real topological vector space, and let (Y, S ) be a real topological vector space with a partial order ≥ induced by a closed and convex cone S, with non-empty interior. Let L (X, Y ) be the space of all continuous linear mappings from X to Y . Let K be a non-empty subset of X , and let G : K → L (X, Y ) be a set-valued function. Then the generalized vector variational inequality problem is to find x0 ∈ K and a continuous linear map A ∈ G(x0 ) such that / −intS for all x ∈ K . A(x − x0 ) ∈ Later on in 1993, Lee et al. [30] studied these generalized vector variational inequalities for Normed spaces. Generalized Vector Variational Inequality Problem for Normed Spaces [30] Let X and Y be two normed spaces, and let K be a non-empty closed and convex

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subset of X . Let T : X → L (X, Y ) be a set-valued mapping, where L (X, Y ) is the space of all continuous linear mappings from X to Y . Let C be a closed, convex and pointed cone in Y with non-empty interior, that is, intC = ∅. Then the generalized vector variational inequality problem is to find x0 ∈ K such that for each x ∈ K , there exists a continuous linear map f ∈ T (x0 ) such that / −intC. f (x − x0 ) ∈ Then, Ansari [2, 3] further generalized the above concept as follow. Generalized Vector Variational-like Inequality Problem [2, 3] Let X and Y be two normed spaces, and let K be a non-empty closed and convex subset of X . Let L (X, Y ) be the space of all continuous linear mappings from X to Y . Let T : X → L (X, Y ) be a set-valued mapping and η : K × K → K be a continuous mapping. Let {C(x) | x ∈ K } be a family of closed, convex and pointed cone in Y with non-empty interior for every x ∈ K , that is, intC(x) = ∅ for all x ∈ K . Then the generalized vector variational-like inequality problem is to find x0 ∈ K such that for each x ∈ K , there exists a continuous linear map f ∈ T (x0 ) such that / −intC(x0 ). f (η(x, x0 )) ∈ After the rigorous study of vector variational inequalities in abstract spaces, Chang and Zhu [6] introduced the notion of variational inequalities for fuzzy mappings. They also discussed the solution sets for these extended variational inequalities. These studies were used to establish the existence of fixed points for fuzzy mappings as well. Fuzzy Extension of Variational Inequality Problem [6] Let X be a reflexive Banach space and Y be a Banach space. Let K be a non-empty closed and convex subset of X and C be a closed, convex and pointed cone in Y with non-empty interior, that is, intC = ∅. Let F : X → F (L (X, Y )) be a fuzzy mapping, where F (L (X, Y )) is the collection of all fuzzy sets on L (X, Y ), the space of all linear continuous mappings. Then fuzzy extension of variational inequality problem is to find x0 ∈ K such that for each x ∈ K , there exists s0 ∈ L (X, Y ) with Fx0 (s0 ) ≥ β; here Fx (u) is the degree of membership of u in Fx , with β ∈ (0, 1] such that s0 (x − x0 ) = −intC. Apart from these, vector variational inequality problems were extended for non-linear analysis as well. In the mid-sixties, Browder [5] provided the initial existence results for the class of non-linear variational inequality problem. He defined the problem on a reflexive Banach space with a monotone non-linear map T from the space X to its dual space X ∗ . After that, the non-linear vector variational inequality problem has been extended and generalized in further course of time. Lie et al. [34], Zhao et al. [42] and Ahmad [1] are a few researchers who extended the work of Browder to a more generalized setup.

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In the subsequent phase, generalized quasi-variational inequalities were also introduced and studied by Hung and others [22–24]. Recently, Irfan et al. [25] introduced a new generalized variational-like problem involving relaxed monotone operators. In 2018, Kim et al. [26] introduced a class of η-generalized operator variational-like inequalities and in the same year, C-pseudo-monotone property for the set-valued mappings was discussed and used to provide a solution set of generalized variational inequality problem by Tavakoli et al. [39]. On the other hand, Farajzadeh, Chen and others [7, 13] studied vector equilibrium problems for multi-functions. This wide range of literature clearly indicates the importance of variational inequality problems gained in recent years. In this chapter, we will discuss some existing results for the existence of solutions to the vector variational inequality problem, generalized vector variational inequality problem and a stronger form of generalized vector variational inequality problem. We also provide some different sets of conditions in the context of topological spaces for the existence of solutions to these problems. This topological approach varies significantly from the rest of the literature. We discuss some examples as well to illustrate the topological results and prove that these conditions are independent of the existing results available in the literature. Topological properties of the solution sets of vector variational and generalized vector variational inequality problems are also discussed.

8.3 Preliminaries In this section, we provide some basic definitions and results on set-valued mappings and function space topologies. We also provide some results and definitions which are used to provide the conditions for solution sets of vector variational inequality and generalized vector variational inequality problems. KKM-Fan theorem plays an important role in providing the existence of solution sets of variational inequalities problems. Definition 1 (KKM-Mapping) [12] Let S be a non-empty subset of a topological vector space X and F : S → X be a set-valued mapping. Then F is said to be a KKM-Map if for every non-empty finite set {x1 , x2 , . . . . . . , xn } of S, we have co{x1 , x2 , . . . . . . , xn } ⊆

n 

F(x j ),

j=1

where co{x1 , x2 , . . . . . . , xn } denotes the convex hull of x1 , x2 , . . . . . . , xn . Now, we provide the KKM-Theorem, which is taken from [12]. Theorem 1 (KKM-Theorem) [12] Let S be a non-empty subset of a topological vector space X . Let F : S → X be a set-valued KKM-mapping such that

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(i) for every x ∈ S, F(x) is a closed subset of X ; (ii) there exists some x0 ∈ S such that F(x0 ) is compact. Then



F(x) = ∅.

x∈S

Following are some basic definitions and results regarding set-valued mappings and function space topology, which we will use frequently throughout this chapter. Definition 2 ([18]) Suppose (X, τ1 ) and (Y, τ2 ) are two topological spaces and F : X → Y is a set-valued map. Then (i) F is called upper semi-continuous (in short, u.s.c.) at a point x ∈ X , if for every open set V in Y such that F(x) ⊆ V , there exists an open set U in X with x ∈ U such that F(U ) ⊆ V ; (ii) F is called lower semi-continuous (in short, l.s.c.) at a point x ∈ X , if for every open set V in Y such that F(x) ∩ V = ∅, there exists an open set U in X with x ∈ U such that for each u ∈ U , F(u) ∩ V = ∅; (iii) F is said to be continuous at x ∈ X if it is both upper semi-continuous and lower semi-continuous at x; (iv) F is said to be continuous (resp. u.s.c., l.s.c.) if it is so at each point of X . Lemma 1 ([17]) Suppose (X, τ1 ) and (Y, τ2 ) are topological spaces and F : X → Y is a set-valued map. Then (i) if F is upper semi-continuous at x ∈ X and F(x) is compact then for every net {xα }α∈D in X and yα ∈ F(xα ) with xα → x and yα → y, we have y ∈ F(x); (ii) F is lower semi-continuous at x ∈ X if and only if for every y ∈ F(x) and every net {xα }α∈D with xα → x, there exists a subnet {xβ }β∈Ω , where Ω is a directed subset of D and a net {yβ }β∈Ω such that yβ ∈ F(xβ ) with yβ → y. Theorem 2 ([18]) Let (X, τ ) and (Y, μ) be two topological spaces. Let F : X → Y be a set-valued map. Then F is lower semi-continuous at x ∈ X if for any net {xn }n∈Δ in X converging to x ∈ X , the image net {F(xn )}n∈Δ converges to F(x). Lemma 2 ([27]) Suppose X and Y are topological spaces and F : X → Y is a setvalued upper semi-continuous function. If F(x) is compact for each x ∈ X , then the image of every compact subset of X under F is compact. Below, we will discuss some results related to function space topology. Definition 3 ([4, 18]) Let (Y, μ1 ) and (Z , μ2 ) be two topological spaces. Let C (Y, Z ) be the space of all continuous mappings from Y to Z . A topology τ on C (Y, Z ) is called admissible, if the evaluation map e : C (Y, Z ) × Y → Z , defined by e( f, y) = f (y), is continuous. Definition 4 ([4]) Let { f n }n∈Δ be a net in C (Y, Z ). Then { f n }n∈Δ is said to continuously converge to f if for each net {ym }m∈σ in Y converging to y, { f n (ym )}(n,m)∈Δ×σ converges to f (y) in Z .

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Theorem 3 ([18]) Let (Y, τ ) and (Z , μ) be two topological spaces. Then topology T on C (Y, Z ), the family of continuous mappings from Y to Z is admissible if and only if for any net { f n }n∈Δ in C (Y, Z ), { f n }n∈Δ converges to f in T implies continuous convergence of { f n }n∈Δ to f . Definition 5 ([28]) Suppose F : X → Y is a set-valued map from X to Y . The graph of F, denoted by G (F), is G (F) = {(x, y) ∈ X × Y | x ∈ X, y ∈ F(x)} . Definition 6 ([9]) Let T : X → L (X, Y ), where L (X, Y ) is the space of  all linear continuous operators from X into Y , be a single-valued map. Let C− = C(x) x∈X

be non-empty, {C(x) | x ∈ K } be a family of closed, pointed and convex cones of Y such that intC(x) = ∅. Then T is said to be C− -monotone if for all x, y ∈ X , we have T (y) − T (x), y − x ∈ C− . Definition 7 ([9]) A mapping T : X → L (X, Y ) is said to be hemi-continuous if for all x, y ∈ X , the mapping t → T (x + t y), y is continuous at 0+ . Definition 8 ([28]) Let X and Y be two Banach spaces, and let K be a non-empty subset of the Banach space X . A set-valued mapping T : X → L (X, Y ) is said to be u - hemi-continuous on K if for any x, y ∈ X and α ∈ [0, 1], the mapping α → (T (x + αz), z) with z = y − x is upper semi-continuous at 0+ . Definition 9 ([28]) Let X and Y be two Banach spaces. Let K be a non-empty subset of X . Let T : K → L (X, Y ) be a set-valued mapping, and let D be a convex cone in Y . Let C : K → Y be a set-valued map such that for each x ∈ K , C(x) is a closed, convex and pointed cone with non-empty interior. Then

(i) T is (D)-monotone on K if for every pair of points x, y ∈ K and for all t ∈ T (x)

and t ∈ T (y), we have

(t − t , y − x) ∈ D. (ii) T is (D)-pseudo-monotone on K if for every pair of points x, y ∈ K and for

all t ∈ T (x) and t ∈ T (y), we have



(t , y − x) ∈ D implies (t , y − x) ∈ D. (iii) T is (C x )-pseudo-monotone on K if for every pair of points x, y ∈ K and for

all t ∈ T (x) and t ∈ T (y), we have



/ −intC(x) implies (t , y − x) ∈ / −intC(x). (t , y − x) ∈

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8.4 On Solutions of Vector Variational Inequality Problems In 1992, Chen proved the existence theorem for solutions to vector variational inequality problems. In his results, he used the notions of C− -monotone and hemicontinuity of T . The result was as follows. Theorem 4 Let X be a reflexive Banach space and Y be a Banach space. Let K ⊆ X be a non-empty bounded, closed and convex subset in X . Let C : K → Y be a setvalued mapping such that, for every x ∈ K , C(x) is a closed, pointed and convex cone C(x) with intC− = ∅, and let T : X → L (X, Y ) be with intC(x) = ∅. Let C− = x∈K

a C− -monotone and hemi-continuous mapping on X . Let W : K → Y be a set-valued map defined as W (x) = Y \ {−intC(x)} such that W is upper semi-continuous on K . Then, the vector variational inequality (VVI) problem is solvable. That is, for every u ∈ K , there exists x0 ∈ K such that / −intC(x0 ). Tx0 (u − x0 ) ∈ But recently, Kumar et al. [29] considered the vector variational inequality problem discussed by Chen and provided the existence of solution sets under a topological framework. They stated this problem for topological vector spaces and provided different sets of conditions which are independent from the conditions given by Chen. The result proved by Kumar et al. is as follows. Theorem 5 Let (X, τ ) and (Y, μ) be two topological vector spaces and C(X, Y ) be the space of all continuous linear mappings from X to Y , equipped with an admissible topology. Let K be a non-empty closed, convex and compact subset of X . Let C : K → Y be a set-valued map such that for every x ∈ X , C(x) is a closed, convex and pointed cone with non-empty interior. Also, let W : K → Y be a setvalued map defined by W (x) = Y \ (−intC(x)) such that the graph of W , G (W ), is a closed set in X × Y . Let T : K → C(X, Y ) be a single-valued continuous mapping. Then the vector variational inequality problem is solvable. That is, for every y ∈ K , there exists x0 ∈ K such that / −intC(x0 ). Tx0 (y − x0 ) ∈ Proof Consider a set-valued map F : K → K defined as / −intC(x)} . F(y) = {x ∈ K | Tx (y − x) ∈ / −intC(x), therefore, y ∈ F(y) and hence the map F As Ty (y − y) = Ty (0) = 0 ∈ is well defined.

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The main proof of the theorem is divided into two parts: (i) F is a KKM-map on K ; Let A = {x1 , x2 , . . . , xn } ⊆ K be any finite subset of K . We will show that n  F(xi ). Let, if possible, there exists some co{x1 , x2 , . . . , xn } ⊆ i=1

x¯ ∈ co{x1 , x2 , . . . , xn } such that x¯ ∈ / where λi ≥ 0 and

n 

n 

F(xi ). Then, we have x¯ =

i=1

n 

λi xi

i=1

λi = 1. As x¯ ∈ / F(xi ) for all xi ∈ A, therefore, we have

i=1

¯ ∈ −intC, for each i = 1, 2, . . . , n. Tx¯ (xi − x) n n   Since −intC is convex and λi ≥ 0 with λi = 1, thus, we have λi (Tx¯ (xi − i=1

x)) ¯ ∈ −intC. Now, consider,

i=1

¯ 0 = Tx¯ (x¯ − x)   n n   λi xi − λi x¯ = Tx¯ i=1

i=1

  n  = Tx¯ λi (xi − x) ¯ i=1

=

n 

λi (Tx¯ (xi − x)). ¯

i=1

Since

n 

λi (Tx¯ (xi − x)) ¯ ∈ −intC(x), ¯ it implies 0 ∈ −intC, which is a con-

i=1

tradiction as C is pointed. Therefore, we have co{x1 , x2 , . . . , xn } ⊆

n 

F(xi ).

i=1

Hence F is a KKM-mapping on K . (ii) F(y) is closed for each y ∈ K . Let {xn } be a net in F(y) ⊆ K such that xn converge to some x ∈ X . As K is a closed set, therefore x ∈ K . / −intC(x). Since xn ∈ F(y), We have to show that x ∈ F(y), that is, Tx (y − x) ∈ / −intC(xn ). Thus, Txn (y − xn ) ∈ therefore, by definition, we have Txn (y − xn ) ∈ W (xn ) and hence we have {xn , Txn (y − xn )} ⊆ G (W ). Now, as xn is a convergent net, which converges to x and T is a continuous mapping therefore, by continuous convergence criteria, we have the image net {Txn } converging to Tx . Also, xn is convergent to x, which implies y − xn converges to y − x only. By the given hypothesis, the topology of C (X, Y ) is given to be admissible. Therefore, we have Txn (y − xn ) converging to Tx (y − x), which implies {xn , Txn (y − xn )} con-

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verges to (x, Tx (y − x)). Since G (W ) is closed, thus (x, Tx (u − x)) ∈ G (W ). / −intC(x). Hence, x ∈ F(y) and F(y) is closed. Therefore Tx (y − x) ∈ Now, for each y ∈ K , F(y) is a closed subset of K . Therefore, F(y) is compact as a closed  can deduce  subset of a compact set K . Thus, by the KKM-Theorem, one F(y) = ∅. Hence, there exists x0 ∈ K such that x0 ∈ F(y), that that y∈K

y∈K

/ −intC(x0 ) for every y ∈ K . Therefore, the vector variational is, Tx0 (y − x0 ) ∈ inequality problem is solvable.  The hypothesis used in the above result is purely topological. Here, one can see that admissibility of function space topology and topological net-theory plays an important role in finding solution sets of the vector variational inequality problem. It is adopted from [29]. Now, we are providing an example to illustrate the above result, and we will show that the conditions used in Theorem 5 are independent of the conditions used in Chen’s paper. Example 1 Let X = l 2 , the set of all square summable sequences, and Y = R, the set of all real numbers. Let K ⊂ X be the Hilbert cube of l 2 . That is, K = {x = {xn }n∈N ∈ K | |xn | ≤ n1 for n ∈ N}. Clearly, K is a closed, convex and compact subset of X . Let C : K → Y be a setvalued map defined by C(x) = R+ ∪ {0}, for every x ∈ K . Then C(x) is a closed, convex and pointed cone with intC(x) = ∅, and −intC(x) = (−∞, 0), for  each x ∈ K . Let T : K → C (X, Y ) be a map defined by Tx (y) = −x, y = − xi yi , where x = {xi } and y = {yi } are in K . First, we show that the induced topology of C (X, Y ) is admissible. It can be easily verified by the fact that if {xn }n∈N converges to x in X and { f n }n∈N is a sequence of linear continuous functions which converges to f in C (X, Y ), then we have  f n (xn ) − f (x) =  f n (xn ) − f n (x) + f n (x) − f (x) ≤  f n (xn ) − f n (x) +  f n (x) − f (x) ≤  f n xn − x +  f n (x) − f (x). Therefore, the sequence { f n (xn )} converges to f (x). Now, we will show that x0 = {− n1 } is a solution to the vector variational inequality problem. Consider x = {xn } in K . Then we have Tx0 (x − x0 ) = −x0 , x − x0   1  1 xn + =− − n n 1 1 xn + = n n > 0,

as |xn | ≤

1 . n

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Therefore, Tx0 (x − x0 ) ∈ / −int C(x0 ). Hence, x0 is a solution for the vector variational inequality problem. Now, we will prove that the T is not C− -monotone. Here, C− = C(x) = [0, ∞) = [0, ∞). x∈K

x∈K

Now, consider T (y) − T (x), y − x = T (y − x), y − x = Ty−x (y − x) =  / C− . Hence, T is not C− -monotone. − (yi − xi )2 ∈ Thus, the topological approach used by Kumar et al. is different from the approach and tools used by Chen. At the same time, Kumar et al. also provide the topological conditions for solution sets for another variant of vector variational inequality problem defined on topological vector spaces, which they named as vector variational inequality problem (I). The result is provided below. Theorem 6 Let (X, τ ) and (Y, μ) be two topological vector spaces. Let C (X, Y ) denote the space of all continuous linear mappings from X to Y, equipped with an admissible topology. Let K ⊆ X be a non-empty closed, convex and compact subset of X . Let C ⊆ Y be a closed, convex and pointed cone with intC = ∅. Further, let T : K → C (X, Y ) be a single-valued continuous mapping. Then the vector variational inequality problem (I) is solvable. That is, for every y ∈ K , there exists an x0 ∈ K such that / −intC. Tx0 (y − x0 ) ∈ Here again, with the help of admissibility and net-theory, Kumar et al. show that the vector variational inequality problem (I) is solvable and if we consider C = [0, ∞) in Example 1, then one can easily verify that the vector variational inequality problem (I) has a solution but the mapping T is not C-monotone. In the following result, we discuss the topological properties of solution sets of vector variational inequality and vector variational inequality problem (I) obtained by Kumar et al. Theorem 7 The solution set for the vector variational inequality problem and vector variational inequality problem (I) obtained in Theorems 5 and in 6 is closed as well as compact. Proof In the proof of Theorem 5, we consider a set-valued map F : K → K defined as F(y) = {x ∈ K | Tx (y − x) ∈ / −intC(x)} . Let S be a solution set of the vector variational inequality problem obtained in F(y). As we proved that F(y) is closed Theorem 5. Therefore, we have S = y∈K

for each y ∈ K , thus, S , being the arbitrary intersection of closed sets, is closed. Also, S is a closed subset of a compact space K , thus S is compact as well. 

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After the topological solution of vector variational inequality problems discussed by Kumar et al., Gupta et al. [19] investigated generalized vector variational inequality problem proposed by Lin, Yang and Yao [33] for topological vector spaces. In the literature, Konnov and Yao [28] have established the existence of solution sets to the generalized vector variational inequality problem with the help of pseudomonotonicity and hemi-continuity. The generalized vector variational inequality problem defined by Lin, Yang and Yao [33] is as follows. Generalized Vector Variational Inequality Problem [33] Let X and Y be real Banach spaces with norms  · x and  ·  y , respectively. Let L (X, Y ) denote the space of all bounded linear mappings from X into Y with the uniform norm. Further, suppose that K is a non-empty closed and convex subset of X and T is a set-valued mapping from K into L (X, Y ). Also let C be a set-valued mapping from K into Y , such that C(x) is a proper closed and convex cone of Y with intC(x) = ∅ for each x ∈ K . Then the generalized vector variational inequality problem (in short, GVVIP) is to find x0 ∈ K such that for each x ∈ K , there exists tx ∈ T (x0 ) satisfying / −intC(x0 ). tx (x − x0 ) ∈ Konnov and Yao provided the following result for solution sets of generalized vector variational inequality problems. Theorem 8 Let X and Y be two real Banach spaces, and let K be a non-empty weakly compact and convex subset of X . Let C : K → Y be a set-valued map such that for each x ∈ K , C(x) is a proper closed and convex cone with non-empty interior. Let W : K → Y be another set-valued map defined as W (x) = Y \ (−intC(x)), such that the graph of W is weakly closed in X × Y . Let T : K → L (X, Y ) be C x pseudo-monotone and u-hemi-continuous on K . Then generalized vector variational inequality problem is solvable. Gupta et al. [19] discussed the same problem on the ground of topological vector spaces and proved that the generalized vector variational inequality problem is solvable under a more general setup. They found that upper semi-continuity with the admissibility of function space topology may be used to show that the generalized vector variational inequality problem is solvable. They also discussed the solution sets of a stronger form of a generalized vector variational problem in the light of lower semi-continuity. They provided the following results. Theorem 9 Suppose X and Y are two topological vector spaces and L (X, Y ) is the space of all continuous linear mappings from the space X to Y equipped with an admissible topology. Let K be a non-empty closed, compact and convex subset of X . Suppose that C : K → Y is a set-valued map such that for every x ∈ K , C(x) is a proper closed, convex and pointed cone with intC(x) = ∅. Further suppose that W : K → Y is also a set-valued map defined by W (x) = Y  (−intC(x)) such that graph of W , G (W ), is a closed set in X × Y . Let T : K → L (X, Y ) be a setvalued upper semi-continuous function such that T (x) is compact for every x ∈ K .

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Then, there exists a solution to the generalized vector variational inequality problem. That is, there exists some x0 ∈ K such that for each x ∈ K , there exists tx ∈ T (x0 ) satisfying / −intC(x0 ). tx (x − x0 ) ∈ Proof Here, we define a set-valued map F : K → K by 

/ −intC(x) . F(y) = x ∈ K | ∃ t y ∈ T (x), t y (y − x) ∈ Here, the map F is well define as for all t ∈ T (y), we have t (y − y) = t (0) = 0 ∈ / −intC(y). Thus, y ∈ F(y). Like the proof of Theorem 5, the proof of this theorem is again divided into two parts: (i) F is a KKM-mapping on K . Similar to the proof as discussed in Theorem 5, (ii) F(y) is closed for each y ∈ K : Let {xα }α∈D be a net in F(y), converging to some y0 ∈ K . Therefore, there exists / −intC(xα ). Now, K is a compact subset some tα ∈ T (xα ) such that tα (y − xα ) ∈ of X , and T is an upper semi-continuous function with every T (x) is compact. Therefore, the image of K under the map T is compact and hence T (K ) is compact by Lemma 2. As {tα }α∈D is a net in T (K ), therefore, there exists a convergent subnet {tαk } which converges to some t0 ∈ T (K ). Now, we consider the subnet {xαk } which converges to y0 and tαk ∈ T (xαk ) such that {tαk } converges to t0 . Then, by Lemma 1, we have t0 ∈ T (y0 ). As the space L (X, Y ) is admissible and tαk is a linearly continuous function, therefore, we have the net {xαk , tαk (y − xαk ) is convergent and converges to {y0 , t0 (y − y0 )}. Also, the graph G (W ) is given to be closed, hence (y0 , t0 (y − y0 )) ∈ G (W ). / −intC(y0 ). Hence y0 ∈ F(y). Thus, t0 (y − y0 ) ∈ W (y0 ), that is, t0 (y − y0 ) ∈ By the KKM-Theorem, we have F(y) = ∅. Hence, there exists some x0 ∈ K such that x0 ∈

 y∈K

y∈K

F(y). That is, for each y ∈ K , there exists t y ∈ T (x0 ) such

/ −intC(x0 ). Hence the result. that t y (y − x0 ) ∈



Now, we provide an example to illustrate the above result. But before that, we provide the following definition and lemma which will be used in the example. Definition 10 [19] Suppose X and Y are two topological spaces and T is a setvalued function from X to Y such that for each x ∈ X , T (x) is a finite set, with T (x) = {x 1 , x 2 , . . . , x m } under some ordering. For any x ∈ X and the net {xn }n∈D in X , we say the net {T (xn )} converges in T (x), if all the nets {xn1 }n∈D , {xn2 }n∈D , . . ., {xnm }n∈D are convergent and their limits belong to the set T (x). Further, if T (x) contains m elements for each x ∈ X , then we say T is a m-valued map.

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In the following lemma, Gupta et al. provided a sufficient condition for upper semi-continuity as well as for lower semi-continuity of a two-valued map using net-theory [19]. Lemma 3 Suppose X and Y are two topological spaces and T : X → Y is a twovalued map. If for any convergent net {xn }n∈D in X , {xn } converging to x implies {T (xn )}n∈D converging in T (x), then T is upper semi-continuous at x. Proof Let x ∈ X and T : X → Y be a two-valued map. For brevity, we write T (x) = {x 1 , x 2 }. Let V be an open set with T (x) ⊆ V . If possible, let for no open set U in X containing x, T (U ) ⊆ V . Then for each open set U containing x, there exists some / V or xu2 ∈ / V . It can be shown that xu ∈ U such that T (xu )  V . Then either xu1 ∈ U , the family of all neighbourhood of x, forms a directed set under the inverse set inclusion relation, that is, U ≥ V if U ⊆ V . Thus, we get a net {xu }u∈U in X . This net converges to x in X . However, the nets {xu1 }u∈U and {xu2 }u∈U are not eventually contained in V , simultaneously. Hence, {T (xu )}u∈U does not converge in T (x). Thus our assumption is wrong. Consequently, T is upper semi-continuous at x.  In the following, we provide an example to illustrate Theorem 9. Example 2 Let X = l 2 , the set of all square summable sequences, and Y = R be the set of all real numbers. Let K ⊂ X be the Hilbert cube of l 2 , that is, x ∈ K if and only if x = {xn }n∈N with |xn | ≤ n1 for n ∈ N. Clearly, K is a convex and compact subset



of X . Let T : K → L(X, Y ) be defined   by T (x) = {tx , tx }, where tx (y) = x, y = xi yi and tx

(y) = −y, x = − xi yi , where x = {xi } is in K and y = {yi } is in i

i

X . T (x) being finite is compact in L (X, Y ). We define a set-valued map C : K → Y by C(x) = R+ ∪ {0}, for every x ∈ K . Then −int(C(x)) = (−∞, 0). The induced topology of L (X, Y ) is admissible, which can be verified easily. T is upper semi-continuous function: Let {x n }n∈N be any convergent net, which converges to x in K . Now, tx n (y) = x n , y which converge to x, y = tx (y), thus {tx n }n∈N converges to tx . Similarly, we have that {tx

n }n∈N converges to tx

. Therefore, {T (x n )}n∈N converges to T (x).  Hence, be Lemma 3, T is upper semi-continuous. Now, consider x0 = − n1 n∈N ∈ K . Then for any y = {yn }n∈N in K , we have  1 1 1  1 yn − =− − yn . Thus, tx

0 (y − x0 ) = −y + x0 , x0  = − − n n n n n n

tx

0 (y − x0 ) ≥ 0, as |yn | ≤ n1 . Therefore, tx

0 (y − x0 ) ∈ / (−∞, 0) = −int(C(x)). Hence, x0 is a solution for the generalized vector variational inequality problem. Now, we will show that T is not C x -pseudo-monotone, which also indicates that the conditions used in Theorem 9 are independent of the result obtained in [28]. T is not C x -pseudo-monotone.



1  and y = n1 n∈N , then x, y ∈ K . Now, consider Consider x = n+1 n∈N

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tx (y − x) = x, y − x  1 1 1 = − n+1 n n+1 n > 0. 

1 1 1 / − < 0. Hence, t (y − x) ∈ n + 1 n n n (−∞, 0), but t y

(y − x) ∈ (−∞, 0). Thus, we have tx ∈ T (x) and t y

∈ T (y) such / −int(C(x)), but t y

(y − x) ∈ −int(C(x)). Hence T is not C x that tx (y − x) ∈ pseudo-monotone. But

t y

(y − x) = x − y, y =

From the following example, one can establish that the result obtained by Gupta et al. is also independent of the result obtained in [33]. Example 3 Suppose X , Y , K and C are defined as in the above example. We define S : K → L (X, Y ) by S(x) = {sx }, where sx (y) = −y, x. Then as explained in the above example, S is upper semi-continuous and x0 = {− n1 }n∈N is a solution to the generalized vector variational inequality problem. S is not generalized

1  C-pseudo-monotone

1  on K . We take x = n+2 and y = − n+1 n∈N . Obviously, x, y ∈ K . Now consider n∈N sx (y − x) = −(y − x), x  1 1 1 + = n+1 n+2 n+2 > 0. But s y (y − x) = −(y − x), y  1 1 1 + =− n+1 n+2 n+1 < 0. / (−∞, 0) but s y (y − x) ∈ (−∞, 0). Thus for x, y ∈ K , there Hence, sx (y − x) ∈ / −int(C(x)) but there exists no element s y ∈ exists sx ∈ S(x) such that sx (y − x) ∈ / −int(C(x)). Hence, S is not generalized C-pseudo-monotone S(y) with s y (y − x) ∈ on K . In the next theorem, we provide another set of conditions for the solution of a stronger form of generalized vector variational inequality problem. The problem is to find some x0 ∈ K such that for each x ∈ K , we have / −intC(x0 ) for all t0 ∈ T (x0 ). t0 (x − x0 ) ∈

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It can be easily seen that every solution of a stronger form of GVVIP is also a solution to the GVVI problem. Theorem 10 Suppose X and Y are two topological vector spaces and L (X, Y ) is the space of all continuous linear mappings from the space X to Y , equipped with an admissible topology. Let K be a non-empty closed, compact and convex subset of X . Suppose that C : K → Y is a set-valued map such that for every x ∈ K , C(x) is a proper closed, convex and pointed cone with intC(x) = ∅. Further suppose that W : K → Y is also a set-valued map defined by W (x) = Y  (−intC(x)) such that graph of W , G (W ), is a closed set in X × Y . Let T : K → L (X, Y ) be a setvalued lower semi-continuous function for every x ∈ X and suppose for each x ∈ K , Ux = {y ∈ K : ∃t ∈ T (x) such that t (y − x) ∈ −intC(x)} is convex. Then, there exists a solution to the stronger form of the generalized vector variational inequality problem. That is, there exists some x0 ∈ K such that for each x ∈ K , / −intC(x0 ) for all t0 ∈ T (x0 ). t0 (x − x0 ) ∈ Proof The proof is almost similar to Theorem 9. Therefore, we are providing a rough sketch of the proof. Here, again, we consider a set-valued map F : K → K defined as F(y) = {x ∈ K | t (y − x) ∈ / −intC(x)} for all t ∈ T (x). The proof of the theorem is divided into two parts: (i) F is a KKM-mapping on K : This part can be proved easily on similar lines, which we discussed in Theorem 9. (ii) F(y) is closed for each y ∈ K : Let {xα }α∈D be a net in F(y), converging to some y0 ∈ K . Therefore, for all / −intC(xα ). Now, T is lower semi-continuous t ∈ T (xα ) we have t (y − xα ) ∈ function, therefore, for t0 ∈ T (y0 ), there exists a subnet {xαk } of {xα } with tαk ∈ T (xαk ) such that the subnet {tαk } converges to t0 , in view of Lemma 1. Since the space L (X, Y ) is admissible and tαk is linearly continuous functions, thus {tαk (y − xαk )} converges to t0 (y − y0 ). Then, proceeding as in Theorem 9, we have y0 ∈ F(y). Now F(y) is closed and K is compact. This  implies that F(y) is a compact subset F(y) = ∅. Hence, there exists some of K . Therefore, by KKM-Theorem, x0 ∈ K such that x0 ∈



y∈K

F(y). That is, for each y ∈ K , and for all t0 ∈ T (x0 )

y∈K

/ −intC(x0 ). Hence the result. we have t0 (y − x0 ) ∈



From Theorems 9 and 10, one can conclude the following corollary. Corollary 1 Suppose X and Y are two topological vector spaces and L (X, Y ) is the space of all continuous linear mappings from the space X to Y , equipped with

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an admissible topology. Let K be a non-empty closed, compact and convex subset of X . Suppose that C : K → Y is a set-valued map such that for every x ∈ K , C(x) is a proper closed, convex and pointed cone with intC(x) = ∅. Further suppose that W : K → Y is also a set-valued map defined by W (x) = Y  (−intC(x)) such that graph of W , G (W ), is a closed set in X × Y . Let T : K → L (X, Y ) be a set-valued continuous function for every x ∈ X . Then, there exists a solution to the generalized vector variational inequality problem (I) (GVVIP (I)), that is, there exists some x0 ∈ K such that for each x ∈ K , there exists tx ∈ T (x0 ) satisfying / −intC(x0 ). tx (x − x0 ) ∈

Conclusion The present review of the latest literature reveals that the topological approaches for vector variational inequality are successful in finding solutions to vector variational inequality problems. It is found that the topological concepts such as closed set, compactness and net-theory are useful tools in these studies. Applications of function space topology is another aspect of these studies. It is expected that similar studies may be carried out on other inequality-related problems.

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13. Farajzadeh, A.P., Wangkeeree, R., Kerdkaew, J.: On the existence of solutions of symmetric vector equilibrium problems via nonlinear scalarization. Bull. Iranian Math. Soc. 45(1), 35–58 (2019) 14. Farajzadeh, A.P., Amini-Harandi, A., O’Regan, D., Agarwal, R.P.: New kinds of generalized variational-like inequality problems in topological vector spaces. Appl. Math. Lett. 22(7), 1126–1129 (2009) 15. Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems, variational inequalities and complementarity problems (Proceeding International School, Erice, 1978), pp. 151–186. Wiley, Chichester (1980) 16. Giannessi, F.: On Minty variational principle. In: New Trends in Mathematical Programming, pp. 93–99, Appl. Optim. 13, Kluwer Academic Publishers, Boston, MA (1998) 17. Göpfert, A., Riahi, H., Tammer, C., et al.: Variational methods in partially ordered spaces. Springer, New York (NY) (2003) 18. Gupta, A., Sarma, R.D.: A study of function space topologies for multifunctions. App. Gen. Topol. 18(2), 331–344 (2017) 19. Gupta, A., Kumar, S., Sarma, R.D., Garg, P.K., Georege, R.: A topological approach to the generalized vector variational inequality problem, presented in International Conference on Mathematical Analysis and its Applications (ICMAA—2019), 14–16 Dec 2019 20. Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966) 21. Al-Homidan, S., Ansari, Q.H.: Generalized Minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144(1), 1–11 (2010) 22. Hung, N.V.: Sensitivity analysis for generalized quasi-variational relation problems in locally G-convex spaces. Fixed Point Theory Appl. 2012(158), 13 (2012) 23. Hung, N.V., Kieu, P.T.: On the existence and essential components of solution sets for systems of generalized quasi-variational relation problems. J. Inequal. Appl. 2014(250), 12 (2014) 24. Hung, N.V., Dai, L.X., Köbis, E., Yao, J.C.: The generic stability of solutions for vector quasiequilibrium problems on Hadamard manifolds. J. Nonlinear Var. Anal. 4, 427–438 (2020) 25. Irfan, S.S., Khan, M.F., Farajzadeh, A.P., Shafie, A.: Generalized variational-like inclusion involving relaxed monotone operators. Adv. Pure Appl. Math. 8(2), 109–119 (2017) 26. Kim, J.K., Khanna, A.K.: Tirth Ram, On η-generalized operator variational-like inequalities. Commun. Optim. Theory 2018, Article ID 14 (2018) 27. Kisielewicz, M.: Differential Inclusions and Optimal Control. Polish Scientific Publishers & Kluwer Academic Publishers (1991) 28. Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997) 29. Kumar, S., Gupta, A., Garg, P.K., Sarma, R.D.: Topological variants of vector variational inequality problem, presented in International Conference on Sustainable Development: Modelling and Optimization (ICSDMO), 16–17 Feb 2021 30. Lee, G.M., Kim, D.S., Lee, B.S., Cho, S.J.: Generalized vector variational inequality and fuzzy extension. Appl. Math. Lett. 6(6), 47–51 (1993) 31. Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34(5), 745–765 (1998) 32. Lee, G.M.: On relations between vector variational inequality and vector optimization problem. In: Progress in Optimization, (Perth), pp. 167–179 (1998) 33. Lin, K.L., Yang, D.P., Yao, J.C.: Generalized vector variational inequalities. J. Optim. Theory Appl. 92(1), 117–125 (1997) 34. Liu, Z., Ume, J.S., Kang, S.M.: Generalized nonlinear variational-like inequalities in reflexive Banach spaces. J. Optim. Theory Appl. 126(1), 157–174 (2005) 35. Mishra, S.K., Wang, S.Y.: Vector variational-like inequalities and non-smooth vector optimization problems. Nonlinear Anal. 64(9), 1939–1945 (2006) 36. Mrowka, S.: On convergence of nets of sets. Fund. Math. 45, 237–246 (1958) 37. Siddiqi, A.H., Ansari, Q.H., Ahmad, R.: On generalized variational-like inequalities. Indian J. Pure Appl. Math. 26(12), 1135–1141 (1995)

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Chapter 9

Ideals and Grills Associated with a Rough Set I. Ibedou, S. E. Abbas, and S. Jafari

Abstract In this Chapter, we introduced the interior and the closure operators with respect to an ideal defined on an approximation space generating an ideal approximation space. The approximation spaces have no relation to the associated Nano topological space. Separation axioms and connectedness in approximation spaces and in ideal approximation spaces are defined and studied. Also, we defined grill separation axioms and grill approximation connectedness with respect to a given grill. All results with ideal are the same with respect to a grill and the converse is true. There is an agreement between the notions ideal and grill. Some examples are given to confirm the introduced implications. Keywords Approximation space · Ideal approximation space · Grill · Grill approximation connectedness · Approximation continuity and ideal approximation continuity

9.1 Introduction In 1982, Prof.Pawlak defined the notion of rough sets [15] referring to the uncertainty of intelligent systems characterized by vague parameters. He gave more theoretical aspects of rough sets and their applications in [16]. An approximation space (X, R) is constructed from a universal finite set of objects and an equivalence relation on I. Ibedou Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt e-mail: [email protected]; [email protected] S. E. Abbas Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt e-mail: [email protected] S. Jafari (B) Mathematical and Physical Science Foundation, Slagelse, Denmark e-mail: [email protected]; [email protected]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_9

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these objects. Rough sets are defined in approximation spaces as a set associated to (X, R) having lower and upper approximation sets, and if the lower and the upper approximation sets are equal, then the set is an exact subset of X and there is no roughness. Many researchers studied the relationship between rough sets and topological spaces. The interested reader can google it on the net and find many interesting papers. Lellis Thivagar et. al in [18] introduced a Nano topological space with respect to a subset A of X which is defined in terms of lower and upper approximations of A. Recently, many researchers have used topological approaches in the study of rough sets and their applications. In [19], the authors gave extensions of Nano topology and some of its applications. The idea of grills on a topological space was first introduced by Choquet [5]. The concept of grills has proved to be a powerful supporting and useful tool like nets and filters, for getting a deeper insight into further studying some topological notions such as proximity spaces, closure spaces, and the theory of compactifications and extension problems of different kinds (see [3, 4, 10, 17] for details). In [17], the authors defined and studied a typical topology associated naturally with the existing topology and a grill on a given topological space. Hatir and Jafari [10] defined new classes of sets and gave a new decomposition of continuity in terms of grills. The notion of ideal in topological spaces was defined and studied in [14], and the local function of some subset in a topological space was defined and studied in [20]. In [11], the authors introduced the notion of an ideal open set in a topological space with respect to an ideal I on X . Many studies have been published based on joining an ideal to a topological space as in [6–9, 12, 13]. Separation axioms with respect to an ideal was given in [1], and the notions of continuity via ideals was given in [2]. In this book Chapter, we introduced the interior and the closure operators in an approximation space generating an approximation topological space with no relation to the associated Nano topological space. Also, we joined the notion of the ideal with the approximation space and defined interior and closure operators with respect to that ideal. The local function of some subset B of X with respect to that ideal is playing a main role in defining the related interior and closure operators. Separation axioms in approximation spaces and in ideal approximation spaces are defined and compared with examples to confirm the implications in between. Connectedness in approximation spaces and in ideal approximation spaces was defined and compared with examples to show the implications between them. The local function of a subset B is defined with respect to a grill on X . The notion of a grill on X is similar to the notion of an ideal, and all results studied with ideal approximation spaces are directly correct in the grill approximation spaces. The agreement between ideal and grill approximation is proved. Approximation continuity and ideal approximation continuity are introduced. Throughout this Chapter, let X be a finite set of objects as a universal set, 2 X denotes all the subsets of X , and R is an equivalence relation on X . Then, the pair (X, R) is said to be an approximation space. The coset [x] of an element x ∈ X is given by [x](y) = R(x, y) ∀y ∈ X. For a set K ⊆ X , the lower L R (K ), upper (U R (K )), and boundary region B R (K ) approximation sets are defined by [18] as follows:

9 Ideals and Grills Associated with a Rough Set

L R (K ) =



169

{[x] : [x] ⊆ K }

(9.1)

{[x] : [x] ∩ K = φ}

(9.2)

x∈X

U R (K ) =

 x∈X

B R (K ) = U R (K ) − L R (K )

(9.3)

L R (A), U R (A) and B R (A) are called lower, upper, and boundary region approximation sets associated with the set A in 2 X and based on the equivalence relation R in an approximation space (X, R). Lemma 1 ([18]) For any sets A, B ∈ 2 X , we have: (1) (2) (3) (4) (5) (6) (7)

L R (A) ⊆ A ⊆ U R (A), L R (φ) = U R (φ) = φ and L R (X ) = U R (X ) = X , A ⊆ B =⇒ L R (A) ⊆ L R (B) and U R (A) ⊆ U R (B), L R (A ∩ B) = L R (A) ∩ L R (B) and U R (A ∪ B) = U R (A) ∪ U R (B), L R (A ∪ B) ⊇ L R (A) ∪ L R (B) and U R (A ∩ B) ⊆ U R (A) ∩ U R (B), (U R (A))c = L R (Ac ) and (L R (A))c = U R (Ac ) U R (L R (A)) = L R (L R (A)) = L R (A) and L R (U R (A)) = U R (U R (A)) = U R (A).

Definition 1 Associated with A ⊆ X in an approximation space (X, R), the interior operator int RA : 2 X → 2 X is given as follows: int RA (B) = L R (A) ∩ L R (B) ∀ B = X

and int RA (X ) = X.

(9.4)

Also, the closure operator cl RA : 2 X → 2 X is given as follows: cl RA (B) = (L R (A))c ∪ U R (B) ∀ B = φ

and cl RA (φ) = φ.

(9.5)

The interior and the closure operators satisfy the following conditions. Lemma 2 The following statements hold: int RA (φ) = cl RA (φ) = φ, int RA (X ) = cl RA (X ) = X , int RA (B) ⊆ B ⊆ cl RA (B) ∀B ∈ 2 X , B ⊆ C =⇒ int RA (B) ⊆ int RA (C), cl RA (B) ⊆ cl RA (C) ∀B, C ∈ 2 X , int RA (B ∩ C) = int RA (B) ∩ int RA (C), int RA (B ∪ C) = int RA (B) ∪ int RA (C) ∀B, C ∈ 2X , (5) cl RA (B ∩ C) = cl RA (B) ∩ cl RA (C), cl RA (B ∪ C) = cl RA (B) ∪ cl RA (C) ∀B, C ∈ 2 X ,  (6) int RA (int RA (B)) = int RA (B) and cl RA (cl RA (B)) = cl RA (B) ∀B ∈ 2 X . (1) (2) (3) (4)

Proof (1) to (5) are obvious. (6) follows from (4), (6), (7) in Lemma 1.

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Note that : int RA (B c ) = (cl RA (B))c , cl RA (B c ) = (int RA (B))c ∀B ∈ 2 X .

(9.6)

These operators int RA and cl RA generate a topology in the approximation space defined by TRA = {B ∈ 2 X : B = int RA (B)} or TRA = {B ∈ 2 X : B c = cl RA (B c )}. (9.7) In [18], a Nano topology τ A was defined in (X, R) as follows: τ A = {φ, X, L R (A), U R (A), B R (A)}. There is no relation between the Nano topology τ A constructed on X and the approx imation topology TRA generated by the interior operator as in Eq. (9.7).

9.2 Ideal Approximation Spaces A non empty collection  of subsets of a non empty set X is said to be an ideal [14] on X if it satisfies the following conditions: (1) If A ⊆ B and B ∈ , then A ∈  for all A, B ∈ 2 X , (2) If A ∈  and B ∈ , then (A ∪ B) ∈  for all A, B ∈ 2 X . In order to exclude the non-proper case where the ideal coincides with the set of all subsets of X , it is generally assumed that X ∈ / . In this case,  is a proper ideal on X . Let 1 and 2 be ideals on X . Then, we have 1 is finer than 2 (2 is coarser than 1 ) if 1 ⊇ 2 . The triple (X, R, ) is called an ideal approximation space. Since  is a non empty collection, then the coarsest ideal is  = {φ}. Definition 2 Let (X, R, ) be an ideal approximation space associated with A ∈ 2 X . Then, the local set  A (B)(R, ) of a set B ∈ 2 X with respect to A is defined by  A (B)(R, ) =

 {G ∈ 2 X : B − G ∈ , cl RA (G) = G}.

(9.8)

We will write briefly  A (B) or  A (B)() instead of  A (B)(R, ). Corollary 1 Let (X, R, ◦ ) be an ideal approximation space associated with A ∈ 2 X where ◦ is the trivial ideal {φ} on X . Then, for each B ∈ 2 X , we have  A (B) = cl RA (B).  Proof Since ◦ = {φ},we obtain  A (B) = {K ∈ 2 X : B − K = φ, cl RA (K ) = K }, that is,  A (B) = {K ∈ 2 X : B ⊆ K , cl RA (K ) = K }. Since B ⊆ cl RA (B), cl RA (cl RA (B)) = cl RA (B), then  A (B) ⊆ cl RA (B). Suppose that cl RA (B)   A (B), then there exists K ∈ 2 X , B ⊆ K , cl RA (K ) = K so that cl RA (B)  K . But B ⊆ K implies that U R (B) ⊆ U R (K ). Thus,

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cl RA (B) = (L R (A))c ∪ U R (B) ⊆ (L R (A))c ∪ U R (K ) = cl RA (K ) = K . But this is a  contradiction and therefore  A (B) = cl RA (B). Proposition 1 Let (X, R) be an ideal approximation space associated with A ∈ 2 X and 1 , 2 are two ideals on X . Then, for each G, H ∈ 2 X (1) (2) (3) (4) (5) (6)

G ⊆ H implies  A (G) ⊆  A (H ). If 1 ⊆ 2 , then  A (G)(1 ) ⊇  A (G)(2 ). int RA ( A (G)) ⊆  A (G) = cl RA ( A (G)) ⊆ cl RA (G).  A (G) =  A ( A (G)).  A (G) ∪  A (H ) =  A (G ∪ H ) and  A (G) ∩  A (H ) ⊇  A (G ∩ H ). If 1 ⊆ 2 , H ∈ 1 , then  A (G) =  A (G ∪ H ).

Proof (1): Suppose that  A (G)   A (H ). There exis W ∈ 2 X with H − W ∈  and cl RA (W ) = W such that  A (G)  W . Since G ⊆ H , then G − W ⊆ H − W and then G − W ∈ , cl RA (W ) = W . Thus,  A (G) ⊆ W , which is a contradiction and hence  A (G) ⊆  A (H ). (2): Suppose that  A (G)(1 )   A (G)(2 ), then there exists W ∈ 2 X , G − W ∈ 1 and cl RA (W ) = W such that W   A (G)(2 ). Since 1 ⊆ 2 , then G − W ∈ 2 and cl RA (W ) = W . Hence,  A (G)(2 ) ⊆ W , which is a contradiction. Thus,  A (G)(1 ) ⊇  A (G)(2 ). (3): It is obvious that int RA ( A (G)) ⊆  A (G) = cl RA ( A (G)) direct. Since  A (G) ⊆ cl RA (G), then  A (G) = cl RA ( A (G)) ⊆ cl RA (G). (4): Let  A ( A (G)) = K , that is,  A (G) − K ∈ , cl RA (K ) = K . Suppose that  A (G) = H  K =  A ( A (G)). Then, G − H ∈ , cl RA (H ) = H , which means that H − K ∈  and G − H ∈ . Thus, G − K ⊆ (G − H ) ∪ (H − K ) ∈ , cl RA (K ) = K , and thus  A (G) = H ⊆ K , which is a contradiction. So,  A (G) ⊆  A ( A (G)). From (3), we have  A (G) =  A ( A (G)) = cl RA ( A ( A (G))) ⊆ cl RA ( A (G)) =  A (G). Hence,  A ( A (G)). (5): By (1),  A (G) ⊆  A (G ∪ H ),  A (H ) ⊆  A (G ∪ H ). Hence,  A (G) ∪  A (H ) ⊆  A (G ∪ H ). Now, suppose that  A (G) ∪  A (H )   A (G ∪ H ). Then, there exist B, W ⊆ X such that G − B ∈ , H − W ∈ , cl RA (B) = B, cl RA (W ) = W with B ∪ W   A (G ∪ H ). But G ∪ H − (B ∪ W ) ∈ , and cl RA (B ∪ W ) = B ∪ W . Hence,  A (G ∪ H ) ⊇ B ∪ W , which is a contradiction. Thus,  A (G) ∪  A (H ) =  A (G ∪ H ).  A (G) ∩  A (H ) ⊇  A (G ∩ H ) by the same token. (6): Obvious.  Definition 3 Let (X, R, ) be an ideal approximation space associated with A ∈ 2 X . Then, for any G ⊆ X , define the operators clA , int A : 2 X → 2 X as follows: (clA )(G) = G ∪  A (G) (int A )(G) = G ∩ ( A (G c )c

∀G ∈ 2 X . ∀G ∈ 2 X .

(9.9) (9.10)

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clA and int A are the closure and the interior operators from 2 X into 2 X associated with a set A and an ideal  in the approximation space (X, R). Corollary 2 If  = ◦ , then

A )(G) = cl A (G) =  (G) and (int A )(G) = int A (G) = ( (G c ))c ∀G ∈ 2 X . (cl A A  R R



Proof It is obvious and it follows from Corollary 1.

Proposition 2 Let (X, R, ) be an ideal approximation space associated with A ∈ 2 X . Then, for any G, H ∈ 2 X , we have (1) int RA (G) ⊆ (int A )(G) ⊆ G ⊆ (clA )(G) ⊆ cl RA (G). (2) clA (G c ) = ((intA )(G))c and int A (G c ) = ((clA )(G))c . (3) (G ∪ H ) = (G) ∪ (H ), (G ∩ H ) = (G) ∩ (H ), {clA , int A }. (4) clA (clA (G)) = (clA )(G) and int A (int A (G)) = (int A )(G). (5) If G ⊆ H , then (G) ⊆ (H ) for all ∈ {clA , int A }.

for all ∈



Proof Clear. These ideal approximation operators than TRA ) on X defined by:

clA ,

int A

generate a topology

ωA

(finer

ωA = {B ∈ 2 X : B = intA (B)} or ωA = {B ∈ 2 X : B c = clA (B c )}. (9.11) Definition 4 Let (X, R, ) be an ideal approximation space associated with A ∈ 2 X . Then, (1) An ideal approximation space (X, R, ) (resp. an approximation space (X, R)) is called an ideal-T0 (resp. T0 ) if for every x = y ∈ A, there exists G ⊆ X with / G or there exists H ⊆ X with x ∈ intA (G) (resp. x ∈ int RA (G)) such that y ∈ / H. y ∈ intA (H ) (resp. y ∈ int RA (H )) such that x ∈ (2) An ideal approximation space (X, R, ) (resp. an approximation space (X, R)) is called an ideal-T1 (resp. T1 ) if for every x = y ∈ A, there exist G, H ⊆ X with x ∈ intA (G), y ∈ int A (H ) (resp. x ∈ int RA (G), y ∈ int RA (H )) such that y∈ / G and x ∈ / H. (3) An ideal approximation space (X, R, ) (resp. an approximation space (X, R)) is called an ideal-T2 (resp. T2 ) if for every x = y ∈ A, there exist G, H ⊆ X with x ∈ intA (G), y ∈ int A (H ) (resp. x ∈ int RA (G), y ∈ int RA (H )) such that G ∩ H = φ. Remark 1 (1) It is clear from int A ⊇ int RA that: T2

T1

T0

ideal − T2

ideal − T1

ideal − T0

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(2) Consider an ideal approximation space (X, R, ) associated with A ∈ 2 X and  = {φ}. Then, the ideal separation axioms ideal-Ti are identical to the separation axioms Ti , i = 0, 1, 2. Example 1 Let X = {a, b, c}, X |R = {{a}, {b, c}}, A = {a, b}. Then, L R (A) = {a}, U R (A) = X, (L R (A))c = {b, c}. That is, L R ({a}) = L R ({a, b}) = L R ({a, c}) = {a}, L R ({b, c}) = {b, c} and L R ({b}) = L R ({c}) = φ, which means that we have int RA ({a}) = int RA ({a, b}) = int RA ({a, c}) = {a} and int RA (H ) = φ for any other proper subset H ⊆ X . Hence, for a = b ∈ A, we can find a set K so that a ∈ / K , and thus (X, R) is an approximation T0 but not approximation int RA (K ) and b ∈ T1 -space or approximation T2 -space. Here, the Nano topology associated with A is given by τ A = {φ, X, {a}, {b, c}}, and thus for b = c ∈ X , we can not find a Nano open set containing b and not containing c or also vice versa. Hence, (X, τ A ) is not even Nano T0 -space. Note that: {b, c} is a Nano open set but int RA ({b, c}) = φ = {b, c}. Example 2 Let X = {a, b, c, d}, X |R = {{a, b}, {c, d}}, A = {a, b}. Then, L R (A) = U R (A) = A = {a, b}, (L R (A))c = {c, d}. Now, int RA (K ) = φ for any K ∈ {φ, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, c, d}, {b, c, d}}, int RA (K ) = {a, b} for any K ∈ {{a, b}, {a, b, c}, {a, b, d}} and int RA (X ) = X . That / K or b ∈ is, for a = b in A, there is no subset K ∈ 2 X such that a ∈ int RA (K ) and b ∈ / K . Thus, (X, R) is not approximation T0 -space, and consequently int RA (K ) and a ∈ not approximation T1 -space or approximation T2 -space. Define an ideal 1 on X as 1 = {φ, {b}, {d}, {b, d}}. Only the sets {φ, X, {c, d}} are satisfying cl RA (K ) = K , and then choosing B = {a, c}, we get  A (B c ) =  A ({b, d}) = φ. Then, B ∩ ( A (B c ))c = B ∩ X = B = int A (B) = {a, c}, which / B. means for a = b in A, there exists a set B = {a, c} such that a ∈ intA (B) and b ∈ Thus, (X, R, 1 ) is an ideal approximation T0 -space. But (all subsets K ∈ {{b}, {b, c}, {b, d}, {b, c, d}} not containing a will not satisfy b ∈ intA (K )). That is, (X, R, 1 ) is not ideal approximation T1 -space or ideal approximation T2 -space. Moreover, if we choose 2 = {φ, {a}, {b}, {a, b}}, then the subset K = {b, c} will satisfy  A (K c ) =  A ({a, d}) = {c, d}, and then int A (K ) = {b}, and thus b ∈ / K . Also, we find a disjoint set H = {a, d} satisfying  A (H c ) = intA (K ) and a ∈ / H.  A ({b, c}) = {c, d}, and then int A (K ) = {a}. Thus, a ∈ int A (H ) and b ∈ It follows that (X, R, 2 ) is an ideal approximation T2 -space. It is also an ideal approximation T1 -space and an ideal approximation T0 -space. Here, the Nano topology associated with A is given by τ A = {φ, X, {a, b}}, and thus for a = b ∈ X , we can not find a Nano open set B ⊆ X such that a ∈ B and b ∈ / B or b ∈ B and a ∈ / B, and thus (X, τ A ) is not Nano Ti -space, i = 0, 1, 2. Recall that a mapping f : (X, R) → (Y, R ∗ ) is said to be an approximation continuous (App-cont.) if int RA ( f −1 (H )) ⊇ f −1 (int BR ∗ (H )) ∀H ⊆ Y. It is equivalent to cl RA ( f −1 (H )) ⊆ f −1 (cl BR ∗ (H )) ∀H ⊆ Y.

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Now with respect to A ⊆ X and B ⊆ Y , let us call a mapping f : (X, R, ) → (Y, R ∗ ) an ideal approximation continuous (ideal App-cont.) provided that int A ( f −1 (H )) ⊇ f −1 (int BR ∗ (H )) ∀H ⊆ Y. It is easily shown that it is equivalent to clA ( f −1 (H )) ⊆ f −1 (cl BR ∗ (H )) ∀H ⊆ Y. Also, let us call f : (X, R) → (Y, R ∗ ) an approximation open (App-open) provided that int BR ∗ ( f (G)) ⊇ f (int RA (G)) ∀G ⊆ X, and let f : (X, R) → (Y, R ∗ , ∗ ) B ( f (G)) ⊇ be an ideal approximation open (ideal App-open) provided that int A f (int R (G)) ∀G ⊆ X. Clearly, every App-cont. (resp. App-open) mapping will be ideal App-cont. (resp. ideal App-open) as well (from (1) in Proposition 2) but not conversely. Theorem 1 Let (X, R), (Y, R ∗ ) be two approximation spaces associated with A ⊆ X , B ⊆ Y , respectively,  an ideal on X and f : (X, R) → (Y, R ∗ ) be an injective App-cont. mapping with f (A) = B. Then, X is an ideal Ti -space if Y is an approximation Ti -space, i = 0, 1, 2. Proof Since x = y in A implies that f (x) = f (y) in B, and by the fact that Y is an approximation T2 -space, then there exist C, D ∈ 2Y with f (x) ∈ int BR ∗ (C), f (y) ∈ int BR ∗ (D) such that C ∩ D = φ, that is, x ∈ f −1 (int BR ∗ (C)), y ∈ f −1 (int BR ∗ (D)). B B (C)), y ∈ f −1 (int  (D)). Since f is App-cont., then x ∈ Then, x ∈ f −1 (int  A A int R ( f −1 (C)), y ∈ int R ( f −1 (D)), and thus x ∈ int A ( f −1 (C)), y ∈ intA ( f −1 (D)). So, there exist G = f −1 (C), K = f −1 (D) with x ∈ intA (G), y ∈ int A (K ) and G ∩ K = φ. Hence, (X, R, ) is an ideal T2 -space. Other cases can be shown in a similar way.  Theorem 2 Let (X, R), (Y, R ∗ ) be two approximation spaces associated with A ⊆ X , B ⊆ Y , respectively, ∗ an ideal on Y and f : (X, R) → (Y, R ∗ ) be a surjective App-open mapping with f −1 (B) = A. Then, Y is an ideal Ti -space if X is an approximation Ti -space, i = 0, 1, 2. Proof Since p = q in B implies that f −1 ( p) = f −1 (q) in A, and since X is an approximation T2 -space, then there exist G, K ∈ 2 X with f −1 ( p) ∈ int RA (G), f −1 (q) ∈ int RA (K ) such that G ∩ K = φ, that is, p ∈ f (int RA (G)), q ∈ f (int RA (K )). Since f is App-open, then, p ∈ int BR ∗ ( f (G)), q ∈ int BR ∗ ( f (K )), and thus p ∈ B B ( f (G)), q ∈ int  ( f (K )). It follows that there exist C = f (G), D = f (K ) int  B B (D) and C ∩ D = φ. Hence, (Y, R ∗ , ∗ ) is an ideal with p ∈ int  (C), q ∈ int   T2 -space. Also, other cases can be shown by the same token. Definition 5 Let (X, R, ) be an ideal approximation space associated with A ⊆ X . Then

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(1) Two sets B, C ∈ 2 X are called ideal approximation separated (ideal App-sep.) (resp. approximation separated (App-sep.)) if clA (B) ∩ C = B ∩ clA (C) = φ (r esp. cl RA (B) ∩ C = B ∩ cl RA (C) = φ). (2) A set G ∈ 2 X is called ideal approximation disconnected (ideal App-disconn.) (resp. approximation disconnected (App-disconn.)) set if there exist ideal Appsep. (resp. App-sep.) sets B, C ∈ 2 X such that B ∪ C = G. A set G is called ideal approximation connected (ideal App-conn.) (resp. approximation connected (App-conn.)) if it is not ideal App-disconn. (resp. Appdisconn.). (3) (X, R, ) is called ideal approximation disconnected (ideal App-disconn.) (resp. approximation disconnected) space if there exist ideal App-sep. (resp. App-sep.) sets B, C ∈ 2 X , such that B ∪ C = X . An ideal approximation space(X, R, ) is called ideal approximation connected (ideal App-conn.) (resp. approximation connected (App-conn.)) if it is not ideal App-disconn. (resp. App-disconn.) space. Remark 2 (1) Any two App-sep. sets B, C in 2 X are ideal App-sep. as well (cl RA (W ) ⊇ clA (W ) ∀W ∈ 2 X ). It means that App-disconn. implies ideal App-disconn. and thus, ideal App-conn. implies App-conn. (2) For  = {φ}, observe that App-conn. and ideal App-conn. are identical. This is an example proving that not any ideal App-disconn. set will be an Appdisconn. set. Example 3 Let X = {a, b, c, d, e}, X |R = {{a, b}, {c}, {d, e}} and A = {a, b, d, e}. Then, L R (A) = U R (A) = {a, b, d, e} and (L R (A))c = {c}. Let B = {a, c}, C = {e}, then U R (B) = {a, b, c}, U R (C) = {d, e}. We have cl RA (C) = U R (C) ∪ (L R (A))c = cl RA (B) = U R (B) ∪ (L R (A))c = {a, b, c} and {c, d, e}. Hence, cl RA (C) ∩ B = {c} = φ. Therefore, B, C are not App-sep. sets in X . In fact, the closure of any subset must contain c, and thus any two subsets of {a, c, e} could not be App-sep. Thus, the set K = {a, c, e} is not App-disconn. at all. Define an ideal  on X as follows:  = {φ, {a}, {e}, {a, e}}. Then,  A (B) = {c},  A (C) = {φ}. Therefore, clA (B) = B ∪  A (B) = {a, c} and clA (C) = C ∪  A (C) = {e}, which means that clA (B) ∩ C = clA (C) ∩ B = φ. Thus, B, C are ideal App-sep. sets. Hence, the subset {a, c, e} is ideal App-disconn. The Nano topology τ A is given by τ A = {φ, X, A}, and hence the Nano closed sets are only {c}, φ, X , which means it is impossible to find two separated sets in (X, τ A ). This means that (X, τ A ) is not Nano disconnected space. Note that a set K = {a, b} satisfying K = int RA (K ) cannot be a Nano open set. Proposition 3 Let (X, R, ) be an ideal approximation space associated with A ⊆ X . Then, the following are equivalent:

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(1) (X, R, ) is ideal App-conn. (2) B ∩ C = φ, intA (B) = B, int A (C) = C and B ∪ C = X imply B = φ or C = φ. (3) B ∩ C = φ, clA (B) = B, clA (C) = C and B ∪ C = X imply B = φ or C = φ. Proof (1) ⇒ (2): Let B, C ∈ 2 X with int A (B) = B, int A (C) = C such that B ∩ C = φ and B ∪ C = X . Then, from (2) in Proposition 2.2, we obtain clA (B) = clA (C c ) = (int A (C))c = C c = B, clA (C) = clA (B c ) = (int A (B))c = B c = C. Hence, clA (B) ∩ C = B ∩ clA (C) = B ∩ C = φ. The sets B, C are ideal App-sep. sets so that B ∪ C = X . But (X, R, ) is ideal App-conn. which implies that B = φ or C = φ. The implications (2) ⇒ (3) and (3) ⇒ (1) are obvious.  Proposition 4 Let (X, R, ) be an ideal approximation space associated with A ⊆ X and let B ∈ 2 X . Then, the following are equivalent: (1) B is an ideal App-conn. set. (2) If C, D are ideal App-sep. sets with B ⊆ (C ∪ D), then B ∩ C = φ or B ∩ D = φ. (3) If C, D are ideal App-sep. sets with B ⊆ (C ∪ D), then B ⊆ C or B ⊆ D. Proof (1) ⇒ (2): Let C, D be ideal App-sep. sets with B ⊆ (C ∪ D). That is, clA (C) ∩ D = clA (D) ∩ C = φ so that B ⊆ (C ∪ D). Since clA (B ∩ C) ∩ (B ∩ D) = clA (B) ∩ clA (C) ∩ (B ∩ D) = clA (B) ∩ B ∩ clA (C) ∩ D = φ, clA (B ∩ D) ∩ (B ∩ C) = clA (B) ∩ clA (D) ∩ (B ∩ C) = clA (B) ∩ B ∩ clA (D) ∩ C = φ. Then, (B ∩ C) and (B ∩ D) are ideal App-sep. sets with B = (B ∩ C) ∪ (B ∩ D). But B is ideal App-conn. which means that B ∩ C = φ or B ∩ D = φ. (2) ⇒ (3): If B ∩ C = φ, B ⊆ (C ∪ D). It follows that B = B ∩ (C ∪ D) = (B ∩ C) ∪ (B ∩ D) = B ∩ D. Thus, B ⊆ D. Also, if B ∩ D = φ, then B ⊆ C. (3) ⇒ (1): Let C, D be ideal App-sep. sets such that B = C ∪ D. Then, by (3), B ⊆ C or B ⊆ D. If B ⊆ C, then D = (C ∪ D) ∩ D = B ∩ D ⊆ C ∩ D ⊆ clA (C) ∩ D = φ. Also, if B ⊆ D, then C = (C ∪ D) ∩ C = B ∩ C ⊆ D ∩ C ⊆ clA (D) ∩ C = φ. Hence, B is an ideal App-conn. set.



Theorem 3 Let (X, R), (Y, R ∗ ) be approximation spaces associated with A ⊆ X and B ∈ 2Y , respectively,  be an ideal on X , and f : (X, R, ) → (Y, R ∗ ) be an

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ideal App-cont. mapping. Then, f (G) ∈ 2Y is an App-conn. set if G is an ideal App-conn. set in X . Proof Let C, D ∈ 2Y be App-sep. sets with f (G) = C ∪ D. That is, cl BR ∗ (C) ∩ D = cl BR ∗ (D) ∩ C = φ. Then, G ⊆ ( f −1 (C) ∪ f −1 (D)), and since f is ideal App-cont., we get that clA ( f −1 (C)) ∩ f −1 (D) ⊆ f −1 (cl BR ∗ (C)) ∩ f −1 (D) = f −1 (cl BR ∗ (C) ∩ D) = f −1 (φ) = φ, and in similar way, we have clA ( f −1 (D)) ∩ f −1 (C) ⊆ f −1 (cl BR ∗ (D)) ∩ f −1 (C) = f −1 (cl BR ∗ (D) ∩ C) = f −1 (φ) = φ. Hence, f −1 (C) and f −1 (D) are ideal App-sep. sets in X so that G ⊆ ( f −1 (C) ∪ f −1 (D)). But from (3) in Proposition 2.4, we obtain G ⊆ f −1 (C) or G ⊆ f −1 (D), which means that f (G) ⊆ C or f (G) ⊆ D. Thus, G is ideal App-conn., and consequently G is an App-conn. set in X , and again by (3) in Proposition 2.4, we conclude that f (G) is App-connected in Y .  The implications in the following diagram are satisfied whenever f is App-cont. G is ideal App − conn.

G is App − conn.

f (G) is ideal App − conn.

f (G) is App − conn.

9.3 Grill Approximation Spaces A collection G of 2 X is called a grill [5] on X if G satisfies the following conditions: (1) A ∈ G and A ⊆ B implies that B ∈ G, (2) A, B ⊆ X and A ∪ B ∈ G implies that A ∈ G or B ∈ G. Remark 3 ([13]) Let X be a non empty set and G ⊆ 2 X . Then, G is a grill on X / G} is an ideal on X . iff (G) = {B ∈ 2 X : B ∈

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Conversely; let X be a non empty set and  ⊆ 2 X . Then,  is an ideal on X iff G() = {B c ∈ 2 X : B ∈ } is a grill on X . Definition 6 Let (X, R) be an approximation space associated with A ∈ 2 X and G a grill on X . Define a mapping  A : 2 X → 2 X as follows:  A (B) =



{G ∈ 2 X : B − G ∈ / G, cl RA (G) = G} for all B ∈ 2 X .

Then, the mapping  A is called the operator associated with A ∈ 2 X in the approximation space (X, R) with respect to the grill G. If G = 2 X − {φ}, we have  A (G) = cl RA (G). The triple (X, R, G) is called a grill approximation space. Proposition 5 Let (X, R, G) be a grill approximation space associated with A ∈ 2 X . Then, for any B, C ∈ 2 X , we have: (1) B ⊆ C implies that  A (B) ⊆  A (C), (2)  A (B ∪ C) =  A (B) ∪  A (C), (3)  A ( A (B)) =  A (B) = cl RA ( A (B)) ⊆ cl RA (B). For a grill approximation space (X, R, G) associated with A ∈ 2 X , define the mapping clA : 2 X → 2 X as follows: clA (G) = G ∪  A (G) ∀G ∈ 2 X .

(9.12)

Also, the mapping int A : 2 X → 2 X is defined as follows: int A (G) = G ∩ ( A (G c ))c ∀G ∈ 2 X .

(9.13)

int RA (G) ⊆ int A (G) ⊆ G ⊆ clA (G) ⊆ cl RA (G) ∀G ∈ 2 X .

(9.14)

Note that

If G = 2 X − {φ}, then (clA )(G) = cl RA (G) =  A (G) and (int A )(G) = int RA (G) = ( A (G c ))c ∀G ∈ 2 X . Lemma 3 Let (X, R) be an approximation space associated with A ∈ 2 X . Then, the following hold: (1) G is a grill on X iff  = 2 X − G is an ideal on X . (2) The operators clA , int A on (X, R, ), where  = 2 X − G, and the operators clA , intA on (X, R, G) are identical. Definition 7 Let (X, R, G) be a grill approximation space associated with A ∈ 2 X . Then

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(1) B ⊆ X is said to be -open if B ⊆ int RA ( A (B)). The complement of -open is said to be -closed. (2) B ⊆ X is said to be preopen if B ⊆ int RA (cl RA (B)). The complement of preopen is said to be preclosed. (3) B ⊆ X is said to be G-preopen if B ⊆ int RA (clA (B)). The complement of G-preopen is said to be G-preclosed. Lemma 4 Let (X, R, G) be a grill approximation space associated with A ∈ 2 X . Then (1) If B ⊆ X is -closed, then B ⊇  A (int RA (B)). (2) If B ⊆ X is preclosed, then B ⊇ cl RA (int RA (B)). (3) If B ⊆ X is G-preclosed, then B ⊇ cl RA (int A (B)). Proof For (1): Suppose that B is -closed, then we have B c ⊆ int RA ( A (B c )) ⊆ int RA (cl RA (B c )) = int RA ((int RA (B))c ) = (cl RA (int RA (B)))c ⊆ A c ( A (int R (B))) .  Therefore,  A (int RA (B)) ⊆ B. It is clear that:  − open( − closed) ⇒ G − preopen(G − preclosed) ⇒ preopen(preclosed). Example 4 Let X = {a, b, c, d, e}, X |R = {{a, b, c}, {d}, {e}} and A = {a, d}. Then, L R (A) = {d}, U R (A) = {a, b, c, d} and (L R (A))c = {a, b, c, e}. For B = {c, d}, then U R (B) = {a, b, c, d}, and then cl RA (B) = U R (B) ∪ (L R (A))c = X , and moreover int RA (cl RA (B)) = X , that is, B ⊆ int RA (cl RA (B)). Thus, B is a preopen set. Define a grill G on X as follows: G = 2 X − {φ, {c}, {d}, {c, d}}. Only the subsets {{a, b, c, e}, φ, X } are satisfying cl RA (H ) = H for some H ⊆ X . Since {c, d} − φ = B−φ ∈ / G, then  A (B) = φ. It means that clA (B) = B ∪  A (B) = B = {c, d}. Then, L R (clA (B)) = {d}, which implies that int RA (clA (B)) = {d}, and thus B  int RA (clA (B)). Hence, B is not G-preopen set. Also, L R ( A (B)) = φ, and hence B  int RA ( A (B)) = φ. This means that B is not -preopen set. Example 5 Let X = {a, b, c, d}, X |R = {{a, b}, {c, d}} and define a grill G as follows G = 2 X − {φ, {c}, {d}, {c, d}}. Let A = {c, d}. Then, L R (A) = U R (A) = {c, d} and (L R (A))c = {a, b}. Let B = {a, c, d}. Then, only the subsets {{a, b}, φ, X } are satisfying cl RA (H ) = H for some H ⊆ X . Since B − φ = B ∈ G, then  A (B) = φ, but  A (B) = {a, b} where B − {a, b} = {c, d} ∈ / G, and thus L R ( A (B)) = {a, b}, which means that B  int RA ( A (B)) = φ. Therefore, B is not -preopen set. Observe that clA (B) = B ∪  A (B) = X , and thus B ⊆ int RA (clA (B)) = X . Hence, B is a G-preopen set.

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Corollary 3 Let (X, R) be an approximation space associated with A ∈ 2 X . Then, B ⊆ X is -open (resp. preopen or G-preopen) set iff B is -open (resp. preopen or -preopen) set in (X, R, ) where  = 2 X − G, and B is -open if B ⊆ int RA ( A (B)). The complement of -open is said to be -closed, B is said to be -preopen if B ⊆ int RA (clA (B)). The complement of -preopen is said to be -preclosed. Definition 8 Let (X, R, G) be a grill approximation space associated with A ∈ 2 X . Then: (1) A grill approximation space (X, R, G) is called a grill-T0 if for every x = y ∈ A, / G or there exists H ⊆ X there exists G ⊆ X with x ∈ intA (G) such that y ∈ / H. with y ∈ int A (H ) such that x ∈ (2) A grill approximation space (X, R, G) is called a grill-T1 if for every x = y ∈ A, / G and there exist G, H ⊆ X with x ∈ intA (G), y ∈ int A (H ) such that y ∈ x∈ / H. (3) A grill approximation space (X, R, G) is called a grill-T2 if for every x = y ∈ A, there exist G, H ⊆ X with x ∈ int A (G), y ∈ int A (H ) such that G ∩ H = φ. Definition 9 Let (X, R, G) be a grill approximation space associated with A ∈ 2 X . Then: (1) Two sets B, C ∈ 2 X are called grill approximation separated (grill App-sep.) if clA (B) ∩ C = B ∩ clA (C) = φ. (2) A set G ∈ 2 X is called grill approximation disconnected (grill App-disconn.) set if there exist grill App-sep. sets B, C ∈ 2 X such that B ∪ C = G. A set G is called grill approximation connected (grill App-conn.) if it is not grill approximation disconnected (grill App-disconn.). (3a) (X, R, G) is called grill approximation disconnected (grill App-disconn.) space if there exist grill App-sep. sets B, C ∈ 2 X , such that B ∪ C = X . (3b) A grill approximation space(X, R, G) is called grill approximation connected (grill App-conn.) if it is not grill approximation disconnected (grill Appdisconn.). Remark 4 It is worth noticing that any two grill App-sep. sets B, C ⊆ X are Appsep. sets as well (from which that cl RA (G) ⊆ clA (G) ∀G ∈ 2 X ). Indeed grill Appdisconn. implies App-disconn., and thus App-conn. implies grill App-conn. Corollary 4 (X, R, G) associated with A ∈ 2 X is not only a grill-Ti approximation space , i = 0, 1, 2, but also a grill App-app-conn. iff (X, R, ) associated with A ∈ 2 X is an ideal-Ti approximation space , i = 0, 1, 2 and ideal App-conn., respectively. Take  = 2 X − G, G = 2 X − .

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9.4 Conclusion Here, we introduced semi connectedness in approximation spaces and in Nano topological spaces. Also, we introduced semi continuity and various types of continuity in a generalized way using the concept of operators in approximation spaces. In future, we will discuss the results in the fuzzy case and elaborate the differences between the ordinary case and the fuzzy case.

References 1. Arenas, F.G., Dontchev, J., Puertas, M.L.: Idealization of some weak separation axioms. Acta Math. Hung. 89(1–2), 47–53 (2000) 2. Aslim, G., Caksu Guler, A., Noiri, T.: On decompositions of continuity and some weaker forms of continuity via idealization. Acta Math. Hung. 109(3), 183–190 (2005) 3. Chattopadhyay, K.C., Njastad, K.C., Thron, W.J.: Extensions of closure spaces. Can. J. Math. 29(6), 1277–1286 (1977) 4. Chattopadhyay, K.C., Njastad, K.C., Thron, W.J.: Merotopic spaces and extensions of closure spaces. Can. J. Math. 35(4), 613–629 (1983) 5. Choquet, G.: Sur les notions de filter et grill. Comptes Rendus Acad. Sci. Paris 224, 171–173 (1947) 6. Dontchev, J., Ganster, M., Rose, D.: Ideal resolvability. Topol. Appl. 93(1), 1–16 (1999) 7. Ekici, E., Noiri, T.: Connectedness in ideal topological spaces. Novi Sad J. Math. 38(2), 65–70 (2008) 8. Hamlett, T.R., Jankovic, ´ D.: Ideals in general topology. Gen. Topol. Appl. 115–125 (1988) 9. Hamlett, T.R., Jankovic, ´ D.: Ideals in topological spaces and the set operatory. Bollettino dell’Unione Matematica Italiana, vol. 7, pp. 863–874 (1990) 10. Hatir, E., Jafari, S.: On some new classes of sets and a new decomposition of continuity via grills. J. Ads. Math. Studies 3(1), 33–40 (2010) 11. Jankovic, ´ D., Hamlet, T.R.: New topologies from old via ideals. Am. Math. Monthly 97, 295– 310 (1990) 12. Jankovic, ´ D., Hamlett, T.R.: Compatible extensions of ideals. Bollettino della Unione Matematica Italiana 7(6), 453–465 (1992) 13. Kandil, A., El-Sheikh, S.A., Abdelhakem, M., Hazza, S.A.: On ideals and grills in topological spaces. South Asian J. Math. 5(6), 233–238 (2015) 14. Kuratowski, K.: Topology. Academic Press, New York (1966) 15. Pawlak, Z.: Rough sets. Int. J. Inf. Comput. Sci. 11, 341–356 (1982) 16. Pawlak, Z.: Rough Sets, Theoritical Aspects of Reasoning About Data, vol. 9. Klwer Academic Publishers, Dordrecht (1991) 17. Roy, B., Mukherjee, M.N.: On a typical topology induced by a grill. Soochow J. Math. 33(4), 771–786 (2007) 18. Thivagar, L., Richard, C.: On nano forms of weakly open sets. Int. J. Math. Stat. Invention 1(1), 31–37 (2013) 19. Thivagar, L., Devi, V.: On Multi-granular nano topology. South East Asian Bull. Math. 40, 875–885 (2016) 20. Vaidyanathaswamy, R.: The localization theory in set topology. Proc. Indian Acad. Sci. 20, 51–61 (1945)

Chapter 10

Filter Versus Ideal on Topological Spaces Jiarul Hoque, Shyamapada Modak, and Santanu Acharjee

Abstract Considering the study of filter and ideal on topological spaces, this chapter will search the equivalent definition of Kuratowski’s local function in terms of a filter. To do this, the big set (resp., small set) and homeomorphism are essential parts. As an application of one of our results, we develop a secret information-sharing scheme in topological cryptography. This new secret-sharing scheme is developed for secret information sharing between two military groups to conduct joint operations on a certain day. Keywords Ultrafilter · Big sets · Small sets · Local function · Homeomorphism · Topological cryptography · Secret information

10.1 Introduction General topology is an area of mathematics which is mainly used to study various properties between two topological spaces under homeomorphism. Although it is an area with classical concepts, one may find generalizations of various concepts of general topology in the recent literature. Filter is one of the main ideas of general topology which has a connection with the ideal of Kuratowski [10]. The definition of ideal was first proposed by Kuratowski [10] in 1933. According to [10], if X is a non-empty set, ℘ (X ) is the family of all subsets of X and I ⊆ ℘ (X ) is a non-empty collection, then I is said to be an ideal of X if it satisfies following two conditions: J. Hoque · S. Modak (B) Department of Mathematics, University of Gour Banga, Malda 732103, West Bengal, India e-mail: [email protected] J. Hoque e-mail: [email protected] S. Acharjee Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_10

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(i) if A ∈ I and B ⊆ A, then B ∈ I ; (ii) if A, B ∈ I , then A ∪ B ∈ I . An ideal I on X is called proper if X ∈ / I. Let τ and I be respectively a topology and an ideal on a non-empty set X . The triplet  X, τ, I is then called an ideal topological space. In an ideal topological space X, τ, I , the function (·)∗ : ℘ (X ) → ℘ (X ) is defined  as follows: for  local  ∗ / I foreveryUx ∈ τ (x) , where τ (x) is A ⊆ X , A I , τ = x ∈ X : Ux ∩ A ∈ the collection of all open sets containing x. A∗ I , τ is simply denoted as A∗I or A∗ . The complement set operator of the set operator (·)∗ is I [13], and it is defined as I (A) = X \ (X \ A)∗ . The set operator C : ℘ (X ) → ℘ (X ) defined by C(A) = A ∪ A∗ makes a closure operator [8, 10], and it is denoted as “Cl ∗ ”, that is Cl ∗ (A) = A ∪ A∗ . This closure operator induces a topology on X , and it is called the ∗-topology [1, 5–7, 11, 12, 15]. This topology is denoted as τ ∗ , and its interior operator for a subset A of X is I nt ∗ (A) = A ∩ I (A) [13]. Recently, Selim et al. [16] and Pal et al. [14] defined a new type of filter, called associated filter, and a new type of local function in terms of filter, where the local function has originally been defined by Kuratowski [10] in terms of ideal. Various aspects related to associated filter have been studied by Selim et al. in [16]. As the associated filter is induced from the ideal, however, the local function in terms of the ideal and the local function in terms of the associated filter are not equivalent. Through this chapter, we solve the problem when these two local functions are equivalent. This chapter will solve a new way to find out a filter from ideal through big sets.

10.2 Big Sets In this section, we introduce J -ideal space, Big set, etc. and study various fundamental properties of them.   Definition 10.1 An ideal topological space X, τ, I is said to be a J -ideal space if it has the property: P :

“A∗ = ∅ for some A ∈ ℘ (X ).”

  Theorem 10.1 For a J -ideal space X, τ, I , I is a proper ideal.   Proof Let X, τ, I be a J -ideal space. If possible, assume that  I = ℘ (X ). Then A∗ = ∅ for all A ∈ ℘ (X ), contradicting the fact that X, τ, I is a J -ideal space. Thus I = ℘ (X ) and hence X ∈ / I . Therefore I is a proper ideal on X .  About the converse of the above theorem, we consider the following example.

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Example 10.1 Let X = N, where N denotes the set of all natural numbers and   τ = ℘ (X ). Consider I = I f the collection of all finite subsets of N . Then I is a proper ideal on X . Clearly, X ∗ = ∅ and hence A∗ = ∅ for all A ∈ ℘ (X ). Thus X, τ, I is not a J -ideal space.   Recall that an ideal I on a topological space X, τ is codense if and only if τ ∩ I = ∅ [4]. When an ideal topological space meets this condition, it is called Hayashi-Samuel space [3]. One of its equivalent conditions is X ∗ = X [4]. It is noticeable that a Hayashi-Samuel space is a J -ideal space but the converse is not true as seen from the following example.          Example 10.2 Let X = a, b, c , τ = ∅, a , X and I = ∅, a . Then X ∗ =     b, c = X . Therefore, X, τ, I is not a Hayashi-Samuel space though it is a J ideal space. Keeping Example 10.1 in mind, we start our main discussion of this chapter. Now, we innovate a new concept by the name of “big set” in J -ideal space instead of the so-called ideal topological space in the following manner:   Definition 10.2 Let X, τ, I be a J -ideal space. A subset A of X is said to be big if B ∗ ⊆ A for all B ∈ ℘ (X ). Throughout this entire chapter, we represent the collection of all big sets of a J -ideal space X, τ, I by Big X, τ . Remark 10.1 It is quite obvious that big sets in a J -ideal space are non-empty. The following examples show that a big set is not always unique in J -ideal space.          Example 10.3 Let X = a, b, c , τ = ∅, a , X and I = ∅, a . Here,  ∗   ∗  ∗  ∗  ∗   ∗   ∅∗ =  a = ∅, b = c = a, b = a, c = b, c =X ∗ = b, c . Thus, A∗ ⊆ b, c ⊆ X for all A ⊆ X . Therefore, b, c and X are big sets. Note that {b, c} is not a dense set.   Example 10.4 Let X = R, τ = ∅, Q, R and I = ℘ (Q), where R and Q respectively denote the set of reals and rationals. Then for any A ⊆ X ,  ∅, if A ∩ (R \ Q) = ∅ A∗ = R \ Q, if A ∩ (R \ Q) = ∅. Thus, A∗ ⊆ R \ Q for all A ⊆ X . Therefore, R \ Q and its all super sets are big sets in R.      Example 10.5 Let X = N and τ = ∅ ∪ [n)  : n ∈ N , where [n) := k ∈ N :  1 k ≥ n . Consider I = A ⊆ N : is finite . Then one can easily check that n+1 n∈A

X ∗ = X . Thus in this J -ideal space, X is the only big set.

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  Theorem 10.2 Let X, τ, I be a J -ideal space and A, B ∈ ℘ (X ). Then ∅ is not a big set; X is always a big set; for A ∈ Big  X, τ , Cl ∗ (H ) ⊆ H ∪ A for all H ∈ ℘ (X ); for A ∈ Big  X, τ , A is closed in τ ∗ -topology;   for A ∈ Big X, τ and A ⊆ B, B ∈ Big X, τ ;  Aα ∈ Big X, τ , where  is an index set; for Aα : α ∈  ⊆ Big X, τ ,    α∈     Aα ∈ Big X, τ ; 7. for Aα : α ∈  ⊆ Big X, τ , α∈   8. τ ∩ I = ∅ if and only if X is the only big set.

1. 2. 3. 4. 5. 6.

Proof (1), (2). They immediately follow from the definition. (3). It is obvious. (4). It follows from the fact Cl ∗ (A) = A ∪ A∗ ⊆ A ∪ A = A. (5). It is obvious.  Aα ⊆ (6), (7). For all B ∈ ℘ (X ), B ∗ ⊆ Aα for each α ∈  and hence B ∗ ⊆ α∈ 



Aα . So Aα and Aα are big sets. α∈ α∈ α∈   (8). It follows from the fact that τ ∩ I = ∅ if and only if X = X ∗ .      From Theorem 10.2, it is clear that Big X, τ  ∪ ∅  is atopology   on X . InExample 10.4, we have seen that Big X, τ ∪ ∅ = ∅ ∪ A ⊆ R : R \ Q ⊆ A is neither weaker nor stronger topology than τ . Remark 10.2 Complement of a big set can’t be a big set again, and it is assured by Theorem 10.2(7).   Lemma 10.1 Let X, τ, I be a J -ideal space. Then       1. Big  X, τ  ∪ ∅ is an indiscrete topology on X if and only if τ ∩ I = ∅ ; 2. Big X, τ ∪ ∅ is a discrete topology on X if and only if X is a singleton set. Proof (1). It follows  from Theorem 10.2(8).      (2). Let A = A ∈ ℘ (X ) : A∗ = ∅ and Big X, τ ∪ ∅ be discrete topology  on X . Pick x, y ∈ X randomly. Then x is a big set that implies A∗ ⊆ x for all    ∗ A ∈ ℘(X) and hence A∗ ⊆ x  for  all A ∈ A . Therefore, A = x ∗for all  A ∈ A . ∗ ⊆ y for all B ∈ ℘ (X ). Therefore, A ⊆ y for all Since y is a bigset, B    A ∈ A and hence x ⊆ y . This yields x = y, where x, y ∈ X were arbitrary. It shows that X is a singleton set.     Converse   Part:   Let X be a singleton set. Then τ = ∅, X and I = ∅ . Clearly, Big X, τ ∪ ∅ is discrete topology on X .        Corollary 10.1 Let X, τ,I  be a J -ideal space. If Big X, τ ∪ ∅ is a discrete topology on X , then I = ∅ . The converse of the above corollary is not true as seen in the following example.

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    Example 10.6 Let X, τ be a topological space  with X ≥ 2 and I = ∅ . By Theorem 10.2(8), X is the only big set. Thus Big X, τ = ∅, X is not the discrete topology on X .   Theorem 10.3 Let X, τ, I be a J -ideal space. A subset A of X is a big set if and only if X \ A ⊆ I (X \ B) for all B ∈ ℘ (X ). Proof It follows from the fact I (X \ B) = X \ B ∗ .



For considering the next results, we intimate about filter [9], P-filter [17] and ultrafilter [9] from the literature. A non-empty collection G of subsets of a non-empty set X is said to be a filter on X if (i) ∅ ∈ / G ; (ii) for G 1 , G 2 ∈ G , G 1 ∩ G 2 ∈ G ; (iii) for A ⊇ B ∈ G , A ∈ G . Suppose a collection P of subsets of a topological space X, τ satisfies P1 ∩ P2 ∈ P and P1 ∪ P2 ∈ P, for all P1 , P2 ∈ P. Then a sub-collection F of nonempty elements of P is said to be a P-filter if (i) for F1 , F2 ∈ F , F1 ∩ F2 ∈ F ; (ii) for F1 ∈ F and F1 ⊆ F2 ∈ P, F2 ∈ F . A filter F on a set X is called an ultrafilter if it is a maximal element in the collection of all filters on X , partially ordered by inclusion, that is, F is an ultrafilter if it is not properly contained in any filter on X . One of the most popular characterizations of the ultrafilter is as follows. Proposition 10.1 ([2, 9]) For a filter F on a set X , the following statements are equivalent: 1. F is an ultrafilter; 2. For any A ⊆ X , either A ∈ F or X \ A ∈ F ; 3. For any A, B ⊆ X, A ∪ B ∈ F if and only if either A ∈ F or B ∈ F .     Theorem 10.4 Let X, τ, I be a J -ideal space. Then Big X, τ forms a filter on X . In fact, it is a P-filter. Proof It follows from (1), (5) and (7) of Theorem 10.2.    The following example shows that the filter Big X, τ (and from now on abbreviated as Fbig ) is not necessarily an ultrafilter.             Example 10.7 Let X = t, u, v, w , τ = ∅, t , u , t, u , X and I = ∅, t .      Then u, v, w and X are the only big sets. Thus Fbig = u, v, w , X . Clearly,     Fbig is not an ultrafilter on X because neither t, u nor its complement v, w are members of Fbig . We shall now show by the following example that Fbig need not be a Selim et al.’s associated filter FI .

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           Example 10.8 Let X = i, j, k, l , τ = ∅, i , j , i, j , X and I = ∅, i ,      ∗  ∗  ∗  ∗  ∗  ∗ k , i, k . Then ∅∗ = i = k = i, k = ∅, l = i, l = k, l =  ∗  ∗  ∗  ∗  ∗  ∗   and j = i, j = j, k = j, l = i, j, k = i, k, l = k, l ∗      ∗  are j, k, l and X . Thus i, j, l = j, k, l = X ∗ = j, k, l . The big sets           Fbig = j, k, l , X . On the other hand, FI = j, l , i, j, l , j, k, l , X . Clearly Fbig = FI .

10.3 Small Sets   Definition 10.3 Let X, τ, I be a J -ideal space. A subset A of X is said to be a small set if X \ A is a big subset of X . We the collection of all small sets in a J -ideal space (X, τ, I ) by  represent  Sma X, τ . Example 10.9 Consider Example 10.4. Here every subset of Q is a small set in R.   Theorem 10.5 Let X, τ, I be a J -ideal space. Then ∅ is always a small set; X is not a small  set;  for A ∈ Sma  X, τ , A = A ∩ I (A), that is, A is open in τ ∗ -topology; for A ∈ Sma X, τ and B  ⊆ A,  B ∈ Sma X, τ ;  Aα ∈ Sma X, τ ; for {Aα : α ∈ } ⊆ Sma X, τ ,   α∈    6. for {Aα : α ∈ } ⊆ Sma X, τ , Aα ∈ Sma X, τ ;

1. 2. 3. 4. 5.

α∈

7. τ ∩ I = {∅} if and only if ∅ is the only small set; 8. A ∈ Sma X, τ if and only if A ⊆ I (X \ E) for all E ∈ ℘ (X ).    It is clear from the above theorem that Sma X, τ  ∪ X forms a topology on        X . In Example 10.3, we see that Sma X, τ ∪ X = ∅, a , X is weaker than       τ whereas in Example 10.4,  Sma  X, τ ∪ X = ℘ (Q) ∪ R is stronger than τ . Thus, the topologies Sma X, τ ∪ {X } and τ are not comparable.   Itis also noticeable that Sma X, τ is an ideal on X but not a  filter as ∅ ∈ Sma X, τ . We denote this  ideal by Isma . In Example 10.8, Isma = ∅, i witnessing that Isma = I , in general. Proof Proofs are obvious and so omitted.





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      Theorem 10.6 For a J -ideal space X, τ, I , Sma X, τ ∩ Big X, τ = ∅.       Proof assume that Sma X, τ ∩ Big X, τ = ∅. Let P ∈ Sma X, τ ∩  If possible,  Big X, τ . Then P is a small as well as big subset of X . Since P is a small set,  X\P is big. Since the intersection of two big sets is a big set, ∅ = P ∩ X \ P is a big set, which leads us towards a contradiction.  Note that the collection Sma(X, τ ) ∪ Big(X, τ ) gives a topology on X .

10.4 Homeomorphism Through this section, we shall try to give an equivalent definition of local function via a filter.   Definition 10.4 Let X, τ, I be an ideal topological space. For a subset A of X , we define (·)∗F I : ℘ (X ) → ℘ (X ) as   / FI for all closed set H not containing x . A ∗F I = x ∈ X : H ∪ A ∈            Example 10.10 Let X = p, q, r, s , τ = ∅, X, p, q , p, r , p , p, q, r              and I = ∅, q , r , q, r . Let A = p , B = q, r and C = p, q, s . Then   A∗F I = s , B ∗F I = X and C ∗F I = ∅.   Theorem 10.7 Let X, τ, I be an ideal topological space. Then A∗F I = (X \ A)∗I for all A ∈ ℘ (X ). Proof Let x ∈ / A∗F I . Then we can find a closed set H not containing x such that H ∪ A ∈ FI . It yields (X \ H ) ∩ (X \ A) ∈ I , where X \ H is an open set containing x. Clearly then x ∈ / (X \ A)∗I . Therefore (X \ A)∗I ⊆ A∗F I . Now, let y ∈ / (X \ A)∗I . Then one can find an open set U containing y such that U ∩ (X \ A) ∈ I . Thus (X \ U ) ∪ A ∈ FI . Since X \ U is a closed set not  containing y, y ∈ / A∗F I . Therefore A∗F I ⊆ (X \ A)∗I .     Lemma 10.2 ([16]) Let f : X, τ → Y, σ be a bijective function. If I is a proper ideal on X , then f (I ) = { f (I ) : I ∈ I } is a proper ideal on Y .     Lemma 10.3 ([16]) Let f : X, τ → Y, σ be a surjective function. If J is a proper ideal on Y , then f −1 (J ) = { f −1 (J ) : J ∈ J } is a proper ideal on X .     Theorem 10.8 Let X, τ, I be an ideal topological space and Y, σ be a topolog∗F I ical )=

space ∗ such that f : X → Y is a homeomorphism. Then for A ⊆ X , f (A f (A) F f (I ) .

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Proof Let y ∈ / f (A∗F I ). Then f −1 (y) ∈ / A∗F I . This shows that there exists a −1 closed set H in X not containing f (y) such that H ∪ A ∈ FI . Thus X   \ (H ∪ A) ∈ I implies Y \ f (H ) ∪ f (A) = Y \ f (H ∪ A) = f X \ (H ∪ A) ∈ f (I ) and it implies f (H ) ∪ f (A) ∈ F f (I ) . Since f (H

) is a∗closed set in Y not ∗ containing y, we must have y ∈ / f (A) F f (I ) . Therefore f (A) F f (I ) ⊆ f (A∗F I ).

∗ set K in Y not containing p Now, let p ∈ / f (A) F f (I ) . So we can find a closed  such that K ∪ f (A) ∈ F . This implies Y \ K ∪ f (A) ∈ f (I ) implies X \ f (I )        f −1 K ∪ f (A) = f −1 Y \ K ∪ f (A) ∈ I . Thus f −1 K ∪ f (A) ∈ FI and hence f −1 (K ) ∪ A ∈ FI . Since f −1 (K ) is a closed set in X not containing f −1 ( p), −1 / A∗F I implying that p ∈ / f (A∗F I ). Therefore f (A∗F I ) ⊆ we

have ∗F f f(I ) ( p) ∈ .  f (A)     Corollary 10.2 Let X, τ, I be an ideal topological space and Y, σ be a topological space such that f : X → Y is a homeomorphism. Then for A ⊆ X ,

∗ 1. f (A∗I ) = f (A) f (I ) ;  2. f I (A) =  f (I ) f (A) .

 Proof (1). Using Theorems 10.7 and 10.8, we have f (X \ A)∗F I = f (X \

∗

∗ ∗ A) F f (I ) implies f (A∗I ) = Y \ f (A) F f (I ) . Therefore f (A∗I ) = f (A) f (I ) . 



∗ (2). By Theorem 10.8, we have f (A∗F I ) = f (A) F f (I ) implies f (X \ A)∗I





∗ f (I ) . This implies f X \ I (A) = Y \  f(I ) f (A) implies = Y \ f (A)  Y\  f I (A) = Y \  f (I ) f (A) . This shows that f I (A) =  f (I ) f (A) .      Theorem 10.9 Let J be an ideal on a topological space Y, σ and X, τ be another topological f : X → Y is a homeomorphism. Then for

∗  space

such that ∗F B ⊆ Y , f −1 B F J = f −1 (B) f −1 (J ) . ∗

∗

Proof Suppose x ∈ / f −1 (B F J ). This gives f (x) ∈ / B F J . It means we can find a closed set F in Y not containing f (x) such  that F ∪ B ∈ FJ . So Y \ (F ∪ B) ∈ J implying that X \ f −1 (F ∪ B) = f −1 Y \ (F ∪ B) ∈ f −1 (J ). This implies f −1 (F ∪ B) ∈ F f −1 (J ) implies f −1 (F) ∪ f −1 (B) ∈ F f −1 (J ) , where f −1 (F) is a ∗F

closed set in X not containing x. So we have x ∈ / f −1 (B) f −1 (J ) . Therefore f −1 ∗F −1 ∗ (B) f (J ) ⊆ f −1 (B F J ).

−1 ∗F Now, let t ∈ / f (B) f −1 (J ) . Then there exists a closed set H in X not containing t such that H ∪ f −1 (B) ∈ F f −1 (J ) . Therefore X \ H ∪ f −1 (B) ∈      f −1 (J ) implies Y \ f H ∪ f −1 (B) = f X \ H ∪ f −1 (B) ∈ J . It shows   that f (H ) ∪ B = f H ∪ f −1 (B) ∈ FJ . Since f (H ) is a closed set in Y not con∗ ∗ ∗ / f −1 (B F J ). Therefore f −1 (B F J ) ⊆ taining f (t) ∈ / B F J and hence t ∈

−1 f(t), ∗F −1 f (J ) .  f (B)     Corollary 10.3 Let J be an ideal on a topological space Y, σ and X, τ be another topological space such that f : X → Y is a homeomorphism. Then for B ⊆ Y,

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∗ −1 ∗J 1. f −1 (B ) = f −1 (B) f (J ) ; 

2. f −1 J (B) =  f −1 (J ) f −1 (B) .

 ∗ Proof (1). Using Theorems 10.7 and 10.9, we have f −1 (X \ B) F J = f −1 (X \ ∗F −1

 ∗F B) f (J ) implies f −1 (B ∗J ) = Y \ f −1 (B) f −1 (J ) . This shows that f −1 (B ∗J ) ∗ f −1 (J )

−1 . = f (B)

∗ 

∗F (2). By Theorem 10.9, we have f −1 B F J = f −1 (B) f −1 (J ) implies f −1 (Y \ 

∗ −1   B)∗J = X \ f −1 (B) f (J ) . This gives f −1 Y \ J (B) =X \  f −1 (J ) f −1 (B) 

  implying that X \ f −1 J (B) = X \  f −1 (J ) f −1 (B) . This shows that f −1 J

  (B) =  f −1 (J ) f −1 (B) .      Theorem 10.10 Let X, τ, I be a J -ideal space and Y, σ be a topological space such that f : X → Y is a homeomorphism. Then Y, σ, f (I ) is also a J ideal space.   Proof Since X, τ, I is a J -ideal space, there exists H ∈ ℘ (X ) such that H ∗I = ∅. Also by Theorem 10.1, I is a proper ideal on X . So f (I ∗ f)(Iis) a proper∗ ideal on = f (H I ) = ∅. Y , by Lemma 10.2. Using Corollary 10.2, we have f (H )   Hence Y, σ, f (I ) is a J -ideal space.      Theorem 10.11 Let Y, σ, J be a J -ideal space and  X, τ be a topological  space such that f : X → Y is homeomorphism. Then X, τ, f −1 (J ) is also a J -ideal space. Proof The proof follows from Theorem 10.1, Lemma 10.3 and Corollary 10.3.      Theorem 10.12 Let X, τ, I be a J -ideal space and Y, σ be a topological space such that f : X → Y is a homeomorphism. Then     1. for A ∈ Big X, τ , f (A) is a big set in Y,σ, f (I ) ;  2. for A ∈ Sma X, τ , f (A) is a small set in Y, σ, f (I ) .

∗ Proof (1). Since A is a big set, f −1 (B) (I ) ⊆ A for all B ∈ ℘ (Y ). By Corollary 10.3, f −1 B ∗ f (I ) ⊆ A. Thus B ∗ f (I ) ⊆ f (A) for all B ∈ ℘ (Y ). Therefore f (A) is a big subset of Y . (2). It is obvious.      Theorem 10.13 Let Y, σ, J be a J -ideal space and X, τ be a topological space such that f : X → Y is a homeomorphism. Then     1. for B ∈ Big Y, σ , f −1 (B) is a big set in X, τ, f −1 (J ) ;  2. for B ∈ Sma Y, σ , f −1 (B) is a small set in X, τ, f −1 (J ) . Proof The proof follows from Corollary 10.2.



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10.5 Topological Cryptography and Applications of Big Set In this section, we show some applications of our results from Big sets in topological cryptography. Topological cryptography is an area of topology which is recently classified by Acharjee [18]. Definition 10.5 ([18]) The area of cryptography which uses topology, topological properties, topological theorems, etc. will be known as “Topological Cryptography”. Here, only topological and related hybrid properties will play central roles in cryptography. The importance of topological cryptography lies in the fact that information sharing needs a secure and secret process. In case of defence matters of any country, secret information-sharing process with other military groups is the main priority of any military group. But it is customary to find that military groups are facing this threat frequently. In 2019, the U.S. Army was hacked by fifty-two hackers within a few weeks [19]. Similar cases have been reported in India also. A group of hackers hacked the phone data of army personnel in Jammu and Kashmir [20] in 2021. “The Times of India” reported the case of hacking their salary details in 2015 [21]. Prior to it, a few Chinese hackers hacked [22] the computer system scores of Indian embassies along with several other classified files. They stole reports of the foreign ministry on India’s policies in West Africa, Russia and West Asia [22]. Moreover, several classified files on security situations in several north-eastern states of India along with Maoist problems [22]. Thus, the secret information-sharing system is not adequate up to that mark as would be expected. In the case of quantum computing, the threats are more than that of a classical computing system of two bits [23]. It is assumed from recent literature that quantum computers and related computing systems will bring threats to security systems of present computing systems of two bits [24]. Thus, threats to secret information of defence may be seen in future. From the above, it can be understood that there are several loopholes in information theory and communication systems. Thus, there is not any harm to develop an alternate process for secure and secret information sharing. The presence of topological aspects almost in every domain of human knowledge has claimed the possible scopes of topology in cryptography but not with algebraic or number theoretical perspectives. Acharjee [18] mainly focused on topological properties, which can be used to create new cryptographical systems. The importance of topological cryptography lies in the following questions. Is it possible to encrypt any information in a topological space (irrespective of general topological or algebraic topological sense)? Can the encrypted information in topological space be deformed? Can the information be decrypted from the deformed topological space by homeomorphism or else? These are some of the questions which open the pavement of topological cryptography. If information can be encrypted and decrypted in a topological space, then it is expected that it may not be easier to hack any secret information too easily. Since

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topological cryptography is in its infancy, a strong theoretical foundation is important for the hybrid structures of topology and information theory or communication systems. In [18], Acharjee proved that ideal I in a topological space has the characteristic to carry secret information in case of a secret communication system. Thus, as an extension of Acharjee [18], we are here to contribute a topological secret informationsharing scheme. The flexibility of these schemes is restored in the fact that topological properties are base of these secret information-sharing schemes. Thus, unlike Shor’s algorithm of quantum computing, which could break the widely used RSA scheme, topological properties in topological cryptography are not assumed to be broken easily. But it is customary for us to look to the future either to prove or disprove the following conjecture. Conjecture 10.1 In topological cryptography [18], if Bob sends an encrypted information A using a topological property TP to Alice and Alice receives topological decrypted information B of the encrypted information A, then it is not possible to decrypt the information A by any third party with any known method of cryptography unless he has full (or partial) information of the property TP. In the above conjecture, the partial information of TP indicates a topological property TP which can be written in a logical sense as TP : P ⇐⇒ Q, where both P and Q are distinct topological properties which are logically equivalent. Otherwise, only TP : P =⇒ Q. Now, we are going to develop a topological secret information scheme for the secret information sharing between two military groups and to conduct possible joint military operations on a certain day. The steps of this secret information scheme are given as follows:   Step 1: Let G 1 and G 2 be two military groups. Let Y, σ, J be a J -ideal topological space  with respect to the set of places Y for the military group G 2 . Let  X, τ, f −1 (J ) be the f −1 (J )-ideal topological space with respect to the set of places X for the military groups G 1 . Step 2: Both the military groups G 1 and G 2 agree on mutual aspects to operating together on a certain day based on the pre-defined homeomorphism. Step 3: Let B ∈ Big Y, σ, J be a big set which is the subset of Y on which military group G 2 will do the military operation. Step 4: Using Theorem 10.13, military group G 1 will receive the set of the places f −1 (B), which is a subset of X . Step 5: If B ∩ f −1 (B) = ∅, then both the military groups will not conduct any military operation on a certain day which are common in both B and f −1 (B). Step 6: If B ∩ f −1 (B) = ∅, then both the military groups will agree to conduct the military operations based on the strategy set B ∩ f −1 (B).    Example 10.11 Consider the J -ideal topological space Y, σ, J with Y =            p,q,        r, s , σ = ∅, p , p, q , p, r , p, q, r , Y and J = ∅, p , r , p, r .  ∗  ∗  ∗  ∗   ∗  ∗  ∗ Then ∅∗ = p = r = p, r = ∅, s = p, s = r, s = p, r, s =

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   ∗  ∗  ∗  ∗  ∗  ∗  ∗ s and q = p, q = q, r = q, s = p, q, r = p, q, s = q, r, s  = X ∗ = q, s . Clearly, Y, σ, J is a J -ideal space. The big sets in this space are             q, s , q, r, s , p, q, s and Y . Thus, Big Y, σ, J = q, s , q, r, s ,    p, q, s , Y .     we consider a topological space X, τ with X = p, q, x, y and τ =   Now,         ∅, y , p, y , q, y , p, q, y , X . Now, we define f : X → Y such that f (y) = p, f ( p) = q, f (q) = s. Then f is a homeomorphism from  = r, f(x)   X to Y .    −1 −1 Then f (J ) = ∅, y , q , q, y . Then Big X, τ, f (J ) = p, x ,        p, q, x , p, x, y , X . Then, X, τ, f −1 (J ) is the f −1 (J )-ideal topological space with respect to the set of places X for the military group G 1 . Let B = {q, r, s} ∈ Big Y, σ, J be the set of places which is considered by the military group G 2 for military operation a certain day. Then, f −1 (B) = { p, q, x} ∈  −1 X, τ, f (J ) is the set of places for the military group G 1 which is obtained by using Theorem 10.13. So, we have B ∩ f −1 (B) = {q}. Thus, both the military groups G 1 and G 2 will conduct military operations jointly in the place “q”. Since Definition 10.3 suggests that the complement of each big set is a small set, thus we may define a new kind of topological secret sharing scheme using small sets and (2) of Theorem 10.13. Since several topology-based secret sharing schemes can be considered as examples from our previous examples, it is customary to leave this task to our readers. However, it is clear that topological cryptography may surely have a long-run impact on cryptography.

10.6 Conclusion In this chapter, we obtained a method for finding a filter (resp., ideal) from ideal topological space via a big (resp., small) set. Moreover, we determined an equivalent definition of Kuratowski’s local function in terms of the associated filter and its homeomorphic image. As applications of our results, we used Theorem 10.13 to develop an example of a new kind of secret information-sharing scheme in topological cryptography [18]. This new secret-sharing scheme was developed for secret information sharing between two military groups to conduct joint operations on a certain day. Thus, we expect that this chapter will open several new directions in topological research related to filter and ideal.

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References 1. Bandhopadhya, C., Modak, S.: A new topology via -operator. Proc. Nat. Acad. Sci. India 76(A), IV, 317–320 (2006) 2. Bourbaki, N.: General Topology, Chapters 1–4. Springer, Berlin (1989) 3. Dontchev, J.: Idealization of Ganster-Reilly decomposition theorems. arXiv:math/9901017v1 [math.GN], 5 Jan (1999) 4. Dontchev, J., Ganster, M., Rose, D.: Ideal resolvability. Topology App. 93, 1–16 (1999) 5. Hamlett, T.R., Jankovic, ´ D.: Ideals in topological spaces and the set operator . Boll. Un. Mat. Ital. 4-B(7), 863–874 (1990) 6. Hashimoto, H.: On the *-topology and its applications. Fund. Math. 91, 5–10 (1976) 7. Hayashi, E.: Topologies defined by local properties. Math. Ann. 156, 205–215 (1964) 8. Jankovic, ´ D., Hamlett, T.R.: New topologies from old via ideals. Am. Math. Monthly 97, 295–310 (1990) 9. Joshi, K.D.: Introduction to General Topology. Wiley (1983) 10. Kuratowski, K.: Topology, vol. I. Academic Press, New York (1966) 11. Modak, S.: Some new topologies on ideal topological spaces. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 82(3), 233–243 (2012) 12. Modak, S., Bandyopadhyay, C.: A note on -operator. Bull. Malays. Math. Sci. Soc. 30(1, 2), 43–48 (2007) 13. Natkaniec, T.: On I-continuity and I-semicontinuity points. Math. Slovaca 36(3), 297–312 (1986) 14. Pal, S.K., Adhikary, N., Samanta, U.: On ideal sequence covering maps. Appl. Gen. Top. 20(2), 363–377 (2019) 15. Samuel, P.: A topology formed from a given topology and ideal. J. London Math. Soc. 10, 409–416 (1975) 16. Selim, Sk., Noiri, T., Modak, S.: Ideals and the associated filters on topological spaces. Eur. Bull. Math. 2(3), 80–85 (2019) 17. Willard, S.: General Topology. Addison-Wesley, Mass, USA (1970) 18. Acharjee, S.: Secret sharing scheme in defense and big data analytics. In: Noise Filtering for Big Data Analytics, pp. 27–46. De Gruyter (2022) 19. Winder, D.: U.S. Army Hacked By 52 Hackers In Five Weeks. https://www.forbes.com/ sites/daveywinder/2020/01/16/us-army-hacked-by-52-hackers-in-five-weeksheres-why/? sh=a0fe7841669f (information was extracted on 12 May 2021) 20. Hackers leak phone data of Army personnel in J & K. https://www.thehindu.com/ news/national/hackers-leak-phone-data-of-army-personnel-in-jk/article33770477.ece (information was extracted on 12 May 2021) 21. Army officers panic as hackers steal secret data. https://timesofindia.indiatimes.com/ india/Army-officers-panic-as-hackers-steal-secret-data/articleshow/46856789.cms (information was extracted on 12 May 2021) 22. Chinese hackers crack Indias top defence secrets. https://www.hindustantimes.com/delhi/ chinese-hackers-crack-india-s-top-defence-secrets/story-YXnaVSQls47oTnv8Q1vYON. html (information was extracted on 12 May 2021) 23. Denning, D.E.: Is quantum computing a cybersecurity threat?. https://www.americanscientist. org/article/is-quantum-computing-a-cybersecurity-threat (information was extracted on 12 May 2021) 24. Lipmen, P.: How quantum computing will transform cybersecurity. https://www. forbes.com/sites/forbestechcouncil/2021/01/04/how-quantum-computing-will-transformcybersecurity/?sh=7ab5341f7d3f (information was extracted on 12 May 2021)

Chapter 11

Fisher Type Set-valued Mappings in b-metric Spaces and an Application to Integral Inclusion Nilakshi Goswami and Nehjamang Haokip

Abstract In this paper, we consider Fisher type set-valued mappings in a b-metric space and derive some fixed point results considering the Pompeiu-Hausdorff metric with respect to the b-metric. A stability result for the fixed point sets is obtained using some topological characteristics of such mappings. An application is also shown for the existence of a solution to an integral inclusion of Fredholm type. Keywords Fixed point · Fisher type contraction · Set-valued mappings · b-metric space

11.1 Introduction The notion of b-metric spaces is a consequence of the generalizations in the underlying spaces of the famous Banach Contraction Principle. This notion was first introduced by Bakhtin [8] and was formally defined by Czerwik [14]. Definition 1 ([14]) Let X be a non-empty set and s ≥ 1 be a given real number. A function d : X × X −→ [0, ∞) is called b-metric if it satisfies the following properties: 1. 2. 3.

d(x, y) = 0 if and only if x = y; d(x, y) = d(y, x); and d(x, z) ≤ s[d(x, y) + d(y, z)], for all x, y, z ∈ X . Then (X, d, s) is called a b-metric space with coefficient s.

In 1998, Fagin and Stockmeyer [16] considered this distance measure (which they called nonlinear elastic matching (NEM)) and remarked that the same measure has N. Goswami Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India N. Haokip (B) Department of Mathematics, Churachandpur College, Manipur 795128, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_11

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been used for trademark shapes in [13] and to measure ice floes in [25]. Later in 2009, Xia [31] used this distance (which he called quasi-metric) to study the optimal transport path between probability measures. The study of b-metric and related results is a current research interest for several researchers. One may refer to [1, 7, 21, 23, 28], etc., for some recent results in b-metric spaces. Considering a metric space (X, d), for a sequence {xn }, limn→∞ xn = x if and only if limn→∞ d(xn , x) = 0. Then the convergence on (X, d) induces the sequential topology τ on X in the sense of Franklin [18]. It is well-known that the topology τd on X generated by the family of all open subsets of (X, d) and the sequential topology τ on (X, d) are coincident. Then limn→∞ xn = x in (X, d) if and only if limn→∞ xn = x in (X, τ ). Definition 2 ([30]) A topological space X is said to be a sequential space if every sequentially open subset of X is open. For convergence in a b-metric space, we have the following definitions. Definition 3 ([4]) Let (X, d, s) be a b-metric space: (i) A sequence {xn } is said to converge to x in (X, d, s) if limn→∞ d(xn , x) = 0, denoted by limn→∞ xn = x. (ii) A sequence {xn } is said to be Cauchy if limn,m→∞ d(xn , xm ) = 0. (iii) (X, d, s) is said to be complete if every Cauchy sequence in (X, d, s) converges in (X, d, s). Thus, for a b-metric space (X, d, s), the sequential topology τ can be derived from the convergence in X as defined in Definition 3. In 2010, Khamshi and Hussain [22] defined a topology on a b-metric space as follows. Definition 4 ([22]) Let (X, d, s) be a b-metric space. A subset G of X is said to be open if for every y ∈ G, there exists r > 0 such that B(y, r ) ⊂ G, where   B(y, r ) = x ∈ X : d(x, y) < r . Khamshi and Hussain showed that if τd denotes the family of all open sets in the sense of the above definition, then τd is a topology on X . Another approach to generating a topology on a b-metric space is the use of a neighbourhood system (refer to [15]). Let (X, d, s) be a b-metric space. Then the family B of all finite intersections of the family   C = B(x, r ) : x ∈ X, r > 0 is a base of a certain topology τ d on X . In [4], the authors showed that in a b-metric space (X, d, s), τ = τd

and

τd ⊂ τ d .

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Unlike a metric, a b-metric need not be continuous as illustrated by the following examples. Example 1 ([20]) Let X = N ∪ {∞} and d : X × X −→ R be defined by ⎧ 0, if m = n, ⎪ ⎪ ⎪ ⎨ 1 − 1  , if one of m, nis even and the other is even or ∞ m n d(m, n) = ⎪ 5, if one of m, n is odd and the other is odd or ∞ ⎪ ⎪ ⎩ 2, otherwise. Then (X, d, s) is a b-metric space with coefficient s = 5/2. Let xn = 2n, n ∈ N. Then, 1 −→ 0 as n → ∞, d(2n, ∞) = 2n i.e., xn −→ ∞, but d(xn , 1) = 2 = 5 = d(∞, 1).   Example 2 ([4]) Let X = 0, 1, 21 , 13 , . . . , n1 , . . . and ⎧ 0, if x = y, ⎪ ⎪ ⎪ ⎨1, if x = y ∈ {0, 1}, 1  d(x, y) = ⎪|x − y|, if x = y ∈ {0} ∪ 2n : n = 1, 2, 3, . . . , ⎪ ⎪ ⎩ 4, otherwise. Then (X, d, s) is a b-metric space with s = 8/3 and as n → ∞,

1 −→ 0, i.e., d 0, 2n

lim

n→∞

1 = 0. 2n

 1 However, limn→∞ d 1, 2n = 4 = 1 = d(1, 0), showing that d is not continuous in each variable. These characteristics of a b-metric make the theory more interesting and practical, and several authors have made significant contributions to this area. The study of setvalued mappings in b-metric spaces and the corresponding fixed point theory with practical applications is an interesting topic of study in this regard. Motivated by this, in this paper, we consider the extension of Fisher’s fixed point theorem to set-valued mappings in b-metric spaces. We derive some stability results from the corresponding fixed point sets. In the last part of the paper, we show an application of our derived result in the Fredholm type integral inclusion. One of the most profound generalizations of the Banach contraction principle is undoubtedly Nadler’s fixed point theorem [26] which generalized the single-valued contraction mapping to a multivalued or set-valued contraction mapping. Nadler’s theorem initiated the development of geometric fixed point theorems for set-valued

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mappings. Many authors have contributed to the fixed point theory of set-valued mappings. One may refer to [5, 6, 24] and the references therein. The following lemma is useful in proving fixed point existence results in b-metric spaces. Lemma 1 ([2]) Let (X, d, s) be a b-metric space and {xn } be a convergent sequence in X with limn→∞ xn = x. Then for all y ∈ X s −1 d(x, y) ≤ lim inf d(xn , y) ≤ lim sup d(xn , y) ≤ sd(x, y). n→∞

n→∞

In 1977, Fisher [17] proved the following fixed point existence theorem. A generalization of this theorem, among many others in the literature, is given in [3]. Theorem 1 ([17]) Let f be a self-mapping on the complete metric space (X, d) such that for all x, y ∈ X d( f x, f y)2 ≤ k1 d(x, f x)d(y, f y) + k2 d(x, f y)d(y, f x) for some real numbers 0 ≤ k1 < 1 and k2 ≥ 0. Then f has a fixed point. In 1980, Pachpatte [27] extended Fisher’s theorem as follows. Theorem 2 Let f be a self-mapping on the complete metric space (X, d) such that for all x, y ∈ X   d( f x, f y)2 ≤ k1 d(x, f x)d(y, f y) + d(x, f y)d(y, f x)   + k2 d(x, f x)d(y, f x) + d(x, f y)d(y, f y) for some k1 , k2 ≥ 0 and k1 + 2k2 < 1. Then f has a unique fixed point in X . The following result is an extension of Lemma 2.1 [19] for set-valued mappings. The proof follows as in [19]. Lemma 2 Let (X, d, s) be a b-metric space and ϕ : X −→ P(X ) be a set-valued mapping. If {xn } is a sequence in X with xn+1 ∈ ϕxn , satisfying d(xn , xn+1 ) ≤ kd(xn−1 , xn ) for all n ∈ N with k ∈ [0, 1), then {xn } is a Cauchy sequence. Pompeiu first introduced the notion of distance between sets in the complex plane in 1905. Later in 1914, Hausdorff studied the notion of set distance (with a small modification of the notion introduced by Pompeiu) in the natural setting of metric spaces. The following is the version in the setting of b-metric space. For a non-empty set X and a b-metric defined on it, let P(X ) and C B(X ) denote the collections of all non-empty subsets and all closed and bounded subsets of X ,

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respectively. Also, let C(X ) denote the collection of all non-empty compact subsets of X . Then the function d H : C B(X ) × C B(X ) −→ [0, ∞) defined by

 d H (A, B) = max sup d(a, B), sup d(A, b) a∈A

b∈B

  for every A, B ∈ C B(X ), where d(a, B) = inf d(a, b) : b ∈ B , is called the Pompeiu-Hausdorff metric [9, 10] or simply Hausdorff metric with respect to d. Further, if (X, d, s) is complete then C B(X ), d H is also complete. Hausdorff also established that these two distances give the same topology, i.e., they are equivalent. There are other notions of set distances not equivalent to the Pompeiu-Hausdorff metric. However, the Pompeiu-Hausdorff metric possesses a remarkable compactness property: If {An } is a sequence of bounded compact sets then there exists a subset A and a subsequence {An k } such that  lim d H An k , A = 0.

k→∞

This property makes the Pompeiu-Hausdorff metric a fundamental notion in the study of the topologies on families of subsets. For more details on the PompeiuHausdorff metric, one may refer to [9] and the references therein. The following is a direct consequence of the above definition which is also mentioned in [10] for metric spaces. Lemma 3 Let (X, d, s) be a b-metric space, A, B ∈ C B(X ) and δ > 0 be a real number. If d H (A, B) ≤ δ and a ∈ A, then there exists b ∈ B with d(a, b) ≤ δ.

11.2 Main Results We begin this section by proving a fixed point existence result which in essence is of Fisher type fixed point theorem extended for set-valued mappings. In the following discussion, the b-metric d need not be continuous. Theorem 3 Let (X, d, s) be a complete b-metric space and ϕ : X −→ C B(X ) be a set-valued mapping such that there exists real numbers k1 ∈ [0, 1) and k2 ≥ 0 satisfying d H (ϕx, ϕy)2 ≤ k1 d(x, ϕx)d(y, ϕy) + k2 d(x, ϕy)d(y, ϕx)

(11.1)

for all x, y ∈ X . Then ϕ has a fixed point. Proof Let x0 ∈ X and x1 ∈ ϕx0 . Without loss of generality, we assume x1 = x0 (for, if x1 = x0 , then x0 ∈ ϕx0 and the proof is complete). Since d(x1 , ϕx1 ) ≤ d H (ϕx0 , ϕx1 ), by (11.1) we have

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d(x1 , ϕx1 )2 ≤ k1 d(x0 , ϕx0 )d(x1 , ϕx1 ) + k2 d(x0 , ϕx1 )d(x1 , ϕx0 ), and d(x1 , ϕx0 ) = 0 implies d(x1 , ϕx1 ) ≤ d H (ϕx0 , ϕx1 ) ≤ k1 d(x0 , ϕx0 ) ≤ k1 d(x0 , x1 ). By Lemma 3, there exists x2 ∈ ϕx1 (with x2 = x1 ) such that d(x1 , x2 ) ≤ k1 d(x0 , x1 ). By (11.1), we again have d H (ϕx1 , ϕx2 )2 ≤ k1 d(x1 , ϕx1 )d(x2 , ϕx2 ) + k2 d(x1 , ϕx2 )d(x2 , ϕx1 ), i.e., d H (ϕx1 , ϕx2 ) ≤ k1 d(x1 , ϕx1 ) ≤ k1 d(x1 , x2 ). Again, by Lemma 3, there exists x3 ∈ ϕx2 (with x3 = x2 ) such that d(x2 , x3 ) ≤ k1 d(x1 , x2 ) ≤ k12 d(x0 , x1 ). Continuing in this way, we construct a sequence {xn } in X , xn+1 ∈ ϕxn such that d(xn , xn+1 ) ≤ k1 d(xn−1 , xn ) ≤ k1n d(x0 , x1 ), for all n ∈ N. By Lemma 2, {xn } is a Cauchy sequence in X which is complete and hence, there exists z ∈ X such that limn→∞ xn = z. Also, since d(xn , ϕz) ≤ sd(xn , z) + sd(z, ϕz), it follows that lim d(xn , ϕz) ≤ sd(z, ϕz).

n→∞

Now,     d(z, ϕz) ≤ s d(z, xn+1 ) + d(xn+1 , ϕz) ≤ s d(z, xn+1 ) + d H (ϕxn , ϕz) implies

 2  2  d(z, ϕz)2 ≤ s 2 d(z, xn+1 ) + 2d(z, xn+1 )d H (ϕxn , ϕz) + d H (ϕxn , ϕz) . But then, for every n ∈ N, we have by (11.1) and Lemma 1,

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 2 d H (ϕxn , ϕz) ≤ k1 d(xn , ϕxn )d(z, ϕz) + k2 d(xn , ϕz)d(z, ϕxn ) ≤ k1 d(xn , xn+1 )d(z, ϕz) + k2 d(xn , ϕz)d(z, xn+1 ) −→ 0,

as n → ∞,

so that, as n → ∞, we have d(z, ϕz)2 = 0,

or,

d(z, ϕz) = 0.

Since ϕz is closed and bounded, z ∈ ϕz as required. Further, if 0 ≤ k2 < 1 then the fixed point in the above theorem is unique. For, if w is another fixed point of ϕ with z = w, then from (11.1) we get d H (ϕz, ϕw)2 ≤ k1 d(z, ϕz)d(w, ϕw) + k2 d(z, ϕw)d(w, ϕz) < d(z, ϕw)d(w, ϕz). But this a contradiction to the fact that d H (ϕz, ϕw) ≥ d(z, ϕw)

and

d H (ϕz, ϕw) ≥ d(ϕz, w).

Example 3 Consider the complete b-metric space (X, d, s), where X = R and d(x, y) = |x − y|2 for all x, y ∈ X with s = 2. Define a set-valued mapping ϕ : X −→ C B(X ) by ϕx = {x} for all x ∈ X . Then ϕ satisfies (11.1) and any point x ∈ X is a fixed point. Remark 1 In the above result, if we consider a single-valued mapping instead, it reduces to Fisher’s fixed point theorem. Theorem 4 Let (X, d, s) be a complete b-metric space and ϕ : X −→ C B(X ) be a set-valued mapping such that there exist non-negative real numbers k1 and k2 with s 2 k1 < 1, satisfying d(ϕx, ϕy)2 ≤ k1 d(x, ϕx)d(y, ϕy) + k2 d(x, ϕy)d(y, ϕx)

(11.2)

  for all x, y ∈ X , where d(A, B) = inf d(a, b) : a ∈ A, b ∈ B . Then ϕ has a fixed point. Proof The proof is similar to that of Theorem 3. Let x0 ∈ X and x1 ∈ ϕx0 . Without loss of generality, we assume x1 = x0 (for, if x1 = x0 , then x0 ∈ ϕx0 and the proof is complete). By triangle inequality and (11.2), we have 2  d(x1 , ϕx1 )2 ≤ s 2 d(x1 , ϕx0 ) + d(ϕx0 , ϕx1 ) ≤ s 2 k1 d(x0 , ϕx0 )d(x1 , ϕx1 ) + s 2 k2 d(x0 , ϕx1 )d(x1 , ϕx0 ),

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and d(x1 , ϕx0 ) = 0 implies d(x1 , ϕx1 ) ≤ s 2 k1 d(x0 , ϕx0 ) ≤ s 2 k1 d(x0 , x1 ). Therefore, there exists x2 ∈ ϕx1 (with x2 = x1 ) such that d(x1 , x2 ) ≤ s 2 k1 d(x0 , x1 ). Again, by triangle inequality and (11.2), we have d(x2 , ϕx2 )2 ≤ s 2 k1 d(x1 , ϕx1 )d(x2 , ϕx2 ) + s 2 k2 d(x1 , ϕx2 )d(x2 , ϕx1 ), and since x2 ∈ ϕx1 ,

d(x2 , ϕx2 ) ≤ s 2 k1 d(x1 , ϕx1 ).

Therefore, there exists x3 ∈ ϕx2 (with x3 = x2 ) such that d(x2 , x3 ) ≤ s 2 k1 d(x1 , x2 ) ≤ (s 2 k1 )2 d(x0 , x1 ). Continuing in this way, we construct a sequence {xn } in X , xn+1 ∈ ϕxn such that d(xn , xn+1 ) ≤ kd(xn−1 , xn ) ≤ k n d(x0 , x1 ), where k = s 2 k1 < 1, for all n ∈ N. By Lemma 2, {xn } is a Cauchy sequence in X which is complete and hence, there exists z ∈ X such that limn→∞ xn = z. Also, since d(xn , ϕz) ≤ sd(xn , z) + sd(z, ϕz), it follows that lim d(xn , ϕz) ≤ sd(z, ϕz).

n→∞

Now,     d(z, ϕz) ≤ s d(z, xn+1 ) + d(xn+1 , ϕz) ≤ s d(z, xn+1 ) + d(ϕxn , ϕz) implies

 2  2  d(z, ϕz)2 ≤ s 2 d(z, xn+1 ) + 2d(z, xn+1 )d(ϕxn , ϕz) + d(ϕxn , ϕz) . But then, for every n ∈ N, we have by (11.2) and Lemma 1 2  d(ϕxn , ϕz) ≤ k1 d(xn , ϕxn )d(z, ϕz) + k2 d(xn , ϕz)d(z, ϕxn ) ≤ k1 d(xn , xn+1 )d(z, ϕz) + k2 d(xn , ϕz)d(z, xn+1 ) −→ 0, as n → ∞,

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so that, as n → ∞, we have d(z, ϕz)2 = 0,

or,

d(z, ϕz) = 0.

Since ϕz is closed and bounded, we have z ∈ ϕz. The result of Theorem 4 is a stronger result compared to Theorem 3 as can be seen from the following example.   Example 4 Let X = 21n : n = 0, 1, 2, . . . ∪ {0} and d(x, y) = |x − y|2 , then (X, d, s) is a complete b-metric space with s = 2. Consider the set-valued mapping ϕ : X −→ C B(X ) defined by  ϕx = If x = 0 and y =

1 , 2n



1 ,1 2n+1

{1},

, x = 21n , n = 0, 1, 2, . . . x = 0.

  n ≥ 0 or vice versa, then ϕ0 = {1}, ϕ2−n = 2−n−1 , 1 .

2 d H (ϕx, ϕy)2 = max sup d(a, ϕ2−n ), sup d(ϕ0, b)

a∈ϕ0

b∈ϕ2−n

= max d(1, ϕ2−n ), sup d(1, b)

2

b∈ϕ2−n

4  = 1 − 2−n−1

and k1 d(0, ϕ0)d(2−n , ϕ2−n ) + k2 d(0, ϕ2−n )d(2−n , ϕ0)  2  2  2 = k1 · 1 · 2−n−1 + k2 2−n−1 1 − 2−n . For any k1 < 1 and k2 ≥ 0, we can find n large enough so that 4 2  2  2   1 − 2−n−1  k1 · 1 · 2−n−1 + k2 2−n−1 1 − 2−n and condition (11.1) is not satisfied. However, for x = 0 and y = 21n , n ≥ 0 (or vice versa),

2  2 d ϕ0, ϕ2−n = inf d(a, b) : a ∈ {0}, b ∈ {2−n−1 , 1} = 0, and for x =

1 2n

and y =

1 , 2m

m > n (or vice versa),

2  2 d ϕ2−n , ϕ2−m = inf d(a, b) : a ∈ {2−n−1 , 1}, b ∈ {2−m−1 , 1} = 0,

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and thus condition (11.2) is satisfied for all x, y ∈ X and any non-negative real numbers k1 and k2 with s 2 k1 < 1. Hence by Theorem 4, ϕ has a fixed point. x = 1 is the fixed point here. Definition 5 A set-valued mapping ϕ : X −→ C B(X ) on a b-metric space (X, d, s) is said to satisfy property (N) if d(x, ϕx) ≤ d(y, ϕy) holds for every y ∈ ϕx, x ∈ X .

  Consider the b-metric space (X, d, s), where X = 2−n : n = 0, 1, 2, . . . and d(x, y) = |x − y|2 . Let ϕ : X −→ C B(X ) be the set-valued mapping defined by ϕx = 2−n−1 , 1 , x = 2−n , n = 0, 1, 2, . . . . Then ϕ satisfies property (N). In the following result, we consider set-valued mappings with the above property. Theorem 5 Let (X, d, s) be a complete b-metric space and ϕ : X −→ C B(X ) be a set-valued mapping satisfying property (N) such that there exist non-negative real numbers k1 , k2 and k3 with k1 + k2 < 1 satisfying d H (ϕx, ϕy)2 ≤ k1 d(x, y)d(y, ϕy) + k2 d(x, ϕx)d(y, ϕy) + k3 d(x, ϕy)d(y, ϕx), (11.3) for all x, y ∈ X . Then ϕ has a fixed point. Proof As in the proof of Theorem 3, for a given x0 ∈ X we can find x1 = x0 in ϕx0 with d(x1 , ϕx1 )2 ≤ d H (ϕx0 , ϕx1 )2 ≤ k1 d(x0 , x1 )d(x0 , ϕx0 ) + k2 d(x0 , ϕx0 )d(x1 , ϕx1 ) + k3 d(x1 , ϕx0 )d(x0 , ϕx1 ) ≤ k1 d(x0 , x1 )d(x1 , ϕx1 ) + k2 d(x0 , ϕx0 )d(x1 , ϕx1 ) ≤ k1 d(x0 , x1 )d(x1 , ϕx1 ) + k2 d(x0 , x1 )d(x1 , ϕx1 ), i.e.,



by property (N)



d(x1 , ϕx1 ) ≤ d H (ϕx0 , ϕx1 ) ≤ (k1 + k2 )d(x0 , x1 ) = k d(x0 , x1 ),

where k = k1 + k2 < 1. By Lemma 3, there exists x2 ∈ ϕx1 (with x2 = x1 ) such that d(x1 , x2 ) ≤ k d(x0 , x1 ). Continuing in this way, as in Theorem 3, we construct a Cauchy sequence {xn } in X with xn+1 ∈ ϕxn and limn→∞ xn = z for some z ∈ X . Also, since d(xn , ϕz) ≤ sd(xn , z) + sd(z, ϕz), it follows that lim d(xn , ϕz) ≤ sd(z, ϕz).

n→∞

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Now,     d(z, ϕz) ≤ s d(z, xn+1 ) + d(xn+1 , ϕz) ≤ s d(z, xn+1 ) + d H (ϕxn , ϕz) implies

 2  2  d(z, ϕz)2 ≤ s 2 d(z, xn+1 ) + 2d(z, xn+1 )d H (ϕxn , ϕz) + d H (ϕxn , ϕz) . But then, for every n ∈ N, we have by (11.3) and Lemma 1 2  d H (ϕxn , ϕz) ≤ k1 d(xn , z)d(xn , ϕxn ) + k2 d(xn , ϕxn )d(z, ϕz) + k3 d(xn , ϕz)d(z, ϕxn ) ≤ k1 d(xn , z)d(xn , xn+1 ) + k2 d(xn , xn+1 )d(z, ϕz) + k3 d(xn , ϕz)d(z, xn+1 ) −→ 0,

as n → ∞,

so that, as n → ∞, d(z, ϕz)2 = 0,

or,

d(z, ϕz) = 0.

This implies z ∈ ϕz since ϕz is closed and bounded. Theorem 6 Let (X, d, s) be a complete b-metric space and ϕ : X −→ C B(X ) be a set-valued mapping such that there exist non-negative real numbers k1 , k2 , k3 and k4 with k1 + 2sk4 < 1 and s 3 k4 < 1, satisfying d H (ϕx, ϕy)2 ≤ k1 d(x, ϕx)d(y, ϕy) + k2 d(x, ϕy)d(y, ϕx) k3 d(x, ϕx)d(y, ϕx) + k4 d(x, ϕy)d(y, ϕy)

(11.4)

for all x, y ∈ X . Then ϕ has a fixed point. Proof Let x0 ∈ X and x1 ∈ ϕx0 . Without loss of generality, we assume x1 = x0 (for, if x1 = x0 , then x0 ∈ ϕx0 and the proof is complete). Since d(x1 , ϕx1 ) ≤ d H (ϕx0 , ϕx1 ), by (11.4) we have d(x1 , ϕx1 )2 ≤ k1 d(x0 , ϕx0 )d(x1 , ϕx1 ) + k4 d(x0 , ϕx1 )d(x1 , ϕx1 ), i.e., d(x1 , ϕx1 ) ≤ k1 d(x0 , ϕx0 ) + k4 d(x0 , ϕx1 )   ≤ k1 d(x0 , x1 ) + sk4 d(x0 , x1 ) + d(x1 , ϕx1 ) , or,

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d(x1 , ϕx1 ) ≤ kd(x0 , x1 ), 1 +sk4 < 1. where k = k1−sk 4 By Lemma 3, there exists x2 ∈ ϕx1 (with x2 = x1 ) such that

d(x1 , x2 ) ≤ kd(x0 , x1 ). Continuing as in Theorem 3, we get a sequence {xn } in X , xn+1 ∈ ϕxn such that d(xn , xn+1 ) ≤ kd(xn−1 , xn ) for all n ∈ N, which by Lemma 2 is a Cauchy sequence. Hence, there exists z ∈ X such that limn→∞ xn = z. Also, since d(xn , ϕz) ≤ sd(xn , z) + sd(z, ϕz), it follows that lim d(xn , ϕz) ≤ sd(z, ϕz).

n→∞

Now,     d(z, ϕz) ≤ s d(z, xn+1 ) + d(xn+1 , ϕz) ≤ s d(z, xn+1 ) + d H (ϕxn , ϕz) implies

 2  2  d(z, ϕz)2 ≤ s 2 d(z, xn+1 ) + 2d(z, xn+1 )d H (ϕxn , ϕz) + d H (ϕxn , ϕz) . But then, for every n ∈ N, we have by (11.4) and Lemma 1, 2  d H (ϕxn , ϕz) ≤ k1 d(xn , ϕxn )d(z, ϕz) + k2 d(xn , ϕz)d(z, ϕxn ) + k3 d(xn , ϕxn )d(z, ϕxn ) + k4 d(xn , ϕz)d(z, ϕz) ≤ k1 d(xn , xn+1 )d(z, ϕz) + k2 d(xn , ϕz)d(z, xn+1 ) + k3 d(xn , xn+1 )d(z, ϕxn ) + k4 d(xn , ϕz)d(z, ϕz) ≤ sk4 d(z, ϕz)2 ,

as n → ∞.

Therefore, as n → ∞, d(z, ϕz)2 ≤ s 3 k4 d(z, ϕz)2 ,

or,

d(z, ϕz) = 0,

since s 3 k4 < 1. This implies z ∈ ϕz as ϕz is closed and bounded.

11.3 Stability of Fixed Point Sets The stability problem of set-valued mappings has been considered by various researchers recently, for instance [11, 12]. In this section, the stability problem related

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to a sequence of set-valued mappings on a b-metric space is considered. In particular, using the uniform convergence of a sequence of Fisher type set-valued mappings, the stability of corresponding fixed point sets is discussed here. The following lemmas are extensions of Lemmas 2.2 and 2.3 of [11] in a b-metric space setting. Lemma 4 ([11]) Let (X, d, s) be a b-metric space and A, B ∈ C(X ), q ≥ 1. Then for every x ∈ A, there exists y ∈ B such that d(x, y) ≤ qd H (A, B). Lemma 5 ([11]) Let A and B be two non-empty compact subsets of a b-metric space (X, d, s) and ϕ : A −→ C(B) be a multivalued mapping and q ≥ 1. Then for a, b ∈ A and x ∈ ϕa, there exists y ∈ ϕb such that d(x, y) ≤ qd H (ϕa, ϕb). Theorem 7 Let (X, d, s) be a complete b-metric space, and ϕ1 , ϕ2 : X −→ C(X ) be two set-valued mappings satisfying (11.1) for some k1 < 1s and k2 ≥ 0. Then  d H F(ϕ1 ), F(ϕ2 ) ≤ mξ,  ∞ (sk1 )i < ∞ and F(ϕi ) are the sets where ξ = supx∈X d H ϕ1 x, ϕ2 x , m = s 2 i=0 of fixed points of ϕi , i = 1, 2. Proof From Theorem 3, F(ϕi ) = ∅, i = 1, 2. Let y0 ∈ F(ϕ1 ), that is, y0 ∈ ϕ1 y0 . Then for q = 1, by Lemma 4 there exists y1 ∈ ϕ2 y0 such that d(y0 , y1 ) ≤ d H (ϕ1 y0 , ϕ2 y0 ).

(11.5)

By Lemma 5, for y1 ∈ ϕ2 y0 there exists y2 ∈ ϕ2 y1 such that d(y1 , y2 ) ≤ d H (ϕ2 y0 , ϕ2 y1 ). As ϕi , i = 1, 2 satisfies (11.1), arguing similarly as in the proof of Theorem 3, we can get a sequence {yn } in X such that for all n ∈ N yn+1 ∈ ϕ2 yn

and

d(yn , yn+1 ) ≤ k1 d(yn−1 , yn ).

(11.6)

As in the proof of Theorem 3, {yn } is a Cauchy sequence in X and hence there exists y ∗ ∈ X such that lim yn = y ∗ . n→∞

Also, y ∗ is a fixed point of ϕ2 , that is, y ∗ ∈ ϕ2 y ∗ . From the definition of ξ and using (11.5), d(y0 , y1 ) ≤ d H (ϕ1 y0 , ϕ2 y0 ) ≤ ξ = sup d H (ϕ1 x, ϕ2 x). x∈X

By the b-metric triangle inequality and (11.6), we have

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  d(y0 , y ∗ ) ≤ s d(y0 , yn+1 ) + d(yn+1 , y ∗ ) ≤

n 

s i+2 d(yi , yi+1 ) + sd(yn+1 , y ∗ )

i=0

≤ s 2 d(y0 , y1 )

n  (sk1 )i + sd(yn+1 , y ∗ ). i=0

Taking the limit as n → ∞ in the above inequality, we have d(y0 , y ∗ ) ≤ s 2 d(y0 , y1 )

∞  (sk1 )i ≤ s 2 ξ i=0

1 = mξ. 1 − sk1

Thus for an arbitrary y0 ∈ F(ϕ1 ), there exists y ∗ ∈ F(ϕ2 ) such that d(y0 , y ∗ ) ≤ mξ. In an analogous manner, we can show that for an arbitrary x0 ∈ F(ϕ2 ), there exists x ∗ ∈ F(ϕ1 ) such that d(x0 , x ∗ ) ≤ mξ . Hence, we conclude that d H F(ϕ1 ), F(ϕ2 ) ≤ mξ . In the following, we use uniform convergence of a sequence of set-valued mappings.  Theorem 8 Let (X, d, s) be a complete b-metric space and ϕn : X −→ C(X ), n ∈  N be a sequence of set-valued mappings, uniformly convergent to a set-valued mapping ϕ : X −→ C(X ). If for every n ∈ N, ϕn satisfies (11.1) for some k1 < s12 and k2 ≥ 0, then the fixed point sets of ϕn are stable, that is,  lim d H F(ϕn ), F(ϕ) = 0.

n→∞

Proof Since ϕn satisfies (11.1) for all n ∈ N, x, y ∈ X , d H (ϕn x, ϕn y)2 ≤ k1 d(x, ϕn x)d(y, ϕn y) + k2 d(x, ϕn y)d(y, ϕn x)     ≤ k1 s d(x, ϕx) + d(ϕx, ϕn x) s d(y, ϕy) + d(ϕy, ϕn y)     + k2 s d(x, ϕy) + d(ϕy, ϕn y) s d(y, ϕx) + d(ϕx, ϕn x) . So, as n → ∞, using the fact that {ϕn } is uniformly convergent to ϕ, we get d H (ϕx, ϕy)2 ≤ s 2 k1 d(x, ϕx)d(y, ϕy) + s 2 k2 d(x, ϕy)d(y, ϕx), showing that ϕ also satisfies (11.1).  Let ξn = supx∈X d H ϕn x, ϕx , n ∈ N. By the uniform convergence of {ϕn } to ϕ on X , we get

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 lim ξn = lim sup d H ϕn x, ϕx = 0.

n→∞

n→∞ x∈X

From Theorem 7, we get for all n ∈ N,  d H F(ϕn ), F(ϕ) ≤ mξn . Therefore,

 lim d H F(ϕn ), F(ϕ) ≤ lim mξn = 0.

n→∞

n→∞

11.4 Application to Fredholm Type Integral Inclusion In this section, following the method used in [29], we prove the existence of the solution to integral inclusion of Fredholm type by applying our result. Let I = [a, b] for some a, b ∈ R and C(I, R) be the space of all real-valued continuous functions on I . Consider the b-metric (with s = 1) on C(I, R) given by   d(x, y) = sup x(t) − y(t) t∈I

for all x, y ∈ C(I, R). We denote by 2clR the family of all non-empty closed subsets of R. Theorem 9 Consider the integral inclusion of Fredholm type b h(t) ∈ f (t) +

 K t, u, h(u) du,

t ∈ [a, b]

(11.7)

a

where f ∈ C(I, R) and K : I × I × R −→ 2clR is such  that for each x ∈ C(I, R), the set-valued mapping K x (., .) where K x (t, u) := K t, u, x(u) , (t, u) ∈ I × I is lower semicontinuous. If for each t ∈ I , there exists l(t, ·) in the Lebesgue space L 1 (I ), such that b supt∈I a l(t, u)du = c with c ∈ [0, 1) and   d K x (t, u), K y (t, u) ≤ l(t, u) sup d x(u), K x (t, u) u∈I

for all t, u ∈ I and x, y ∈ C(I, R), then the integral inclusion (11.7) has a solution in C(I, R). C(I,R) defined by Proof Consider the set-valued mapping ϕ : C(I, R) −→ 2cl

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b     ϕx = h ∈ C(I, R)h(t) ∈ f (t) + K t, u, x(u) du, t ∈ I

a

for x ∈ C(I, R). Then as in [29], it can be shown that ϕx is non-empty and closed. Let x, y ∈ C(I, R) be arbitrary but fixed. If x ∈ ϕx, then (11.7) has a solution in C(I, R) trivially. We take x ∈ / ϕx. Let h ∈ ϕx. Then there exists k x (t, u) ∈ K x (t, u), (t, u) ∈ I × I such that b h(t) = f (t) +

k x (t, u)du. a

Also, by hypothesis,   d K x (t, u), K h (t, u) ≤ l(t, u) sup d h(u), K h (t, u) u∈I

so that there exists γ (t, u) ∈ K h (t, u) such that    k x (t, u) − γ (t, u) ≤ l(t, u) sup d h(u), K h (t, u) u∈I

for all (t, u) ∈ I × I . Considering the set-valued mapping N (., .) defined by

    N (t, u) = K y (t, u) ∩ α ∈ R : k x (t, u) − α  ≤ l(t, u)d x, ϕx for all (t, u) ∈ I × I , since N is a lower semicontinuous mapping, we get a continuous mapping k y : I × I −→ R such that k y (t, u) ∈ N (t, u) for all t, u ∈ I and b γ (t) = f (t) +

k y (t, u)du ∈ f (t) + a

Therefore,

b K y (t, u)du. a

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    d ϕx, ϕy ≤ d h, γ = sup h(t) − γ (t) t∈I   b        = sup  k x (t, u) − k y (t, u) du  t∈I   a

b ≤ sup t∈I

a

b ≤ sup t∈I

  k x (t, u) − k y (t, u)du  l(t, u)d x, ϕx du

a

≤ cd(x, ϕx). the role of x(t) and y(t), and repeating the same procedure, we get  Interchanging d ϕx, ϕy ≤ cd(y, ϕy) and hence,  2 d ϕx, ϕy ≤ c12 d(x, ϕx)d(y, ϕy) ≤ c12 d(x, ϕx)d(y, ϕy) + c2 d(x, ϕy)d(y, ϕx), where c1 = c and c2 ≥ 0. Hence from Theorem 4, ϕ has a fixed point and the proof is complete.

Conclusion We have obtained fixed point results for Fisher type contractive set-valued mapping defined on a b-metric space and the existence of a solution to an integral inclusion of Fredholm type is also obtained as an application. The stability result for the fixed point sets of a sequence of Fisher type contractive set-valued mappings uniformly convergent to a set-valued mapping is also obtained. As remarked earlier, Theorem 4 is stronger than Theorem 3 though the distance d used in Theorem 4 does not define a metric in C B(X ). In this context, the question arises: Does there exist a metric on C B(X ) for which a stronger result to that of Theorem 3 can be obtained? Finally, the validity of the results obtained in this paper may also be investigated in extended b-metric spaces.

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Chapter 12

Topological Aspects of Granular Computing Santanu Acharjee, Amlanjyoti Oza, and Upashana Gogoi

Abstract Granular computing is an area of theoretical computer science which is highly useful in nature. Recently, several new directions of granular computing have been explored with crisp set, fuzzy set, rough set and related hybrid structures. Here, we try to explore various ideas related to granularity from the perspectives of general topology, binary relations and crisp sets only. Our intention is to attract the attention of topologists, who work with crisp sets, to work in granular computing and thus, we restrict ourselves only to crisp set-based granular computing. At last, we discuss some feasible ideas from biology and microscopy, which may inspire the experts of granular computing to develop new theories based on crisp sets and realities of nature. Keywords Granular computing · Binary granulation · Algebraic quotient space · Microscopy · Neuroscience

12.1 Introduction Granular computing is an emerging research field which provides multidisciplinary approaches to many areas of mathematics, computer science, etc. It is an umbrella term to cover any theory that makes use of granules in problem solving. It was first introduced by Lotfi A. Zadeh [10] in 1979 under the name of ‘information granularity’. Granular computing is applicable on both humans and machines. Here, the main ideas are human inspired, that is, they can be extracted from human thinking and S. Acharjee · A. Oza (B) Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India e-mail: [email protected] S. Acharjee e-mail: [email protected] U. Gogoi Department of Mathematics, Morigaon College, Morigaon 782105, Assam, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_12

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problem solving which later on can be implemented on machines to perform problem solving, computing and information processing [15]. Although it is a new term in the field of research, it has a large variety of applications in several fields such as artificial intelligence [13], databases [14], divide and conquer [5], interval computing [14], machine learning [14] and cluster analysis [14]. We can consider granular computing at two different levels, namely as structured thinking and structured problem solving [6]. Granular computing focuses on structured thinking that concerns the conceptual level at multiple levels of granularity. At the application level, such conceptual thinking is being applied to those which deal with structured problem solving. In this chapter, we discuss some theories and applications related to granular computing after defining some basic concepts related to it from the perspective of topology. According to Zadeh [1], “information granulation involves partitioning a class of objects (points) into granules, with a granule being a clump of objects (points) which are drawn together by indistinguishability, similarity or functionality”. In MerriamWebster’s dictionary [11], a granule is defined as “a small particle; especially, one of the numerous particles forming a larger unit”. From these two definitions, we can conclude that any small particle of a universe may be considered as a granule, which may be a subset, subclass or cluster of the whole universe. On the other hand, granulation actually means the process of making granules from a universe by decomposing it to small particles. According to Zadeh [6], “granulation involves a decomposition of whole into parts. Conversely, organization involves an integration of parts into whole.” From this definition, we may conclude that there are two operations, namely decomposition and organization in the process of granulation. On the other hand, a granular structure is a collection of granules which are interconnected to each other. In the case of problem solving, a granule may be considered as a unit which represents a particular representation of some information. The size of a granule may be considered as the amount of data that can be inserted inside it, for example, for a small amount of data to be processed we shall consider a smaller granule that is interconnected with the others in the granular structure. That is why a granular structure may be considered as a tree of structured understanding and information processing. Pictorially, we can consider a granular structure [13] as shown in Fig. 12.1. The number of granules in the lower level of the structure is smaller and more in numbers relative to the granules at higher level, but they are interconnected. At a particular level of the structure, all the granules have the same characteristics. In this way, a multilevel hierarchy provides a strategy to level-wise information processing.

12.2 Granular Computing in Binary Relations In recent years, granular computing has been studied by many researchers from various points of views based on different concepts and models. Among these models, the one which is based on binary neighborhood system (BNS) was formulated from Zadeh’s informal definition of ‘information granulation’ [1]. The granules are noth-

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Fig. 12.1 Pictorial view of a granular structure

ing but clumps of objects which are drawn together by ‘indistinguishability’, ‘similarity’ and ‘functionality’. Mathematically, the words ‘indistinguishability’, ‘similarity’ or ‘functionality’ mentioned in Zadeh’s definition of information granulation [1] can be expressed in terms of binary relations. The phrase ‘drawn together’ implies the existence of symmetry in the clumps. Naturally, this kind of symmetry does not always exist. To avoid this implication, Lin [2] rephrased the definition of Zadeh [1] as follows. “Information granulation is a collection of granules, with a granule being a clump of data (in data space) which are drawn towards the center object(s) (in object space) by indistinguishability, similarity or functionality.” From the definition, we notice that there does not exist any constraint on the number of clumps an object can be associated with. In a crisp sense, this kind of disorderly collection of granules leads to the notion of neighborhood system (NS) [2]. To put this simply, we consider an object to be a point p and a granule associated with p is a neighborhood of p. The family NS( p) of neighborhoods associated with p is called a neighborhood system of p. NS( p) can be empty, finite or infinite. If for every p, there exists at most one member in NS( p), then it defines a binary relation uniquely and such a system is called a binary neighborhood system (BNS) [4]. If the binary relation is an equivalence relation, then the neighborhood system is a rough set

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system (RS). If the NS satisfies the axioms of topology, then it is called topological neighborhood system (TNS). In other words, BNS is a simple generalization of rough sets and classical topology. In granular computing, neighborhood and neighborhood system are called granule and granular structure, respectively. Granulation in the real world does not always represent an equivalence relation. For example, the relation ‘father of’ does not refer to an equivalence relation. Thus, it was necessary to generalize the theory of equivalence relations (in terms of rough set theory) to the theory of general binary relations (in terms of binary neighborhood system). Furthermore, the notion of partitions (geometric equivalence of an equivalence relation) does not allow overlapping among the granules. But, when we talk about real world problems, overlapping is inevitable. Since a binary granulation is defined by a binary relation, it does not form a partition; thus there does not exist any restriction on overlapping among the granules.

12.2.1 Mathematical Structure of Binary Granulation In this section, we are going to discuss the notion of BNS or equivalently binary granulation based on two universes [2, 3]. For this, we consider the object space V and the data space U . Let B ⊆ V × U be a binary relation. For each p ∈ V , we associate a binary subset B p ⊆ U , where B p = {u ∈ U | p Bu}. Here, B p consists of all elements u that are related to p by the binary relation B. The map B : V → 2U or the collection {B p } is referred to as a (right) binary neighborhood system (BNS) for V on U . Similar to (right) BNS, we can also define (left) BNS. Then, B p is termed as a neighborhood of p. If B p is an empty set, we will say p has no neighborhood or p has an empty neighborhood. There is a possibility that different points may have the same neighborhood, i.e., B p = Bq for p = q and p, q ∈ V . The set of all q, where B p = Bq , is called the center C p of B p [5]. It is clear that a binary relation induces a binary neighborhood system and vice versa. The binary relation B can be defined by Zadeh’s ‘indistinguishability’, ‘similarity’ or ‘functionality’. If p is the center object, then B p is the clump of data drawn toward p by the binary relation B. The map from the object space to the power set of data space is also called a binary granulation (BG). Thus, a binary neighborhood system (BNS), a binary granulation (BG) and a binary relation (BR) are equivalent [5]. In many real-life applications, we need to consider several clumps at the same time. In such cases, we get a family of binary relations {B j | j ∈ ,  is an index set}. Each B j induces a neighborhood B j ( p) at each point p of the object space. Thus, we get a family of subsets of U which are p-oriented and the collection {B j ( p)} is referred to as the neighborhood system (NS) of p. Thus, a neighborhood system (NS) is a map U

N S : V → 22 ; p → N S( p) = {B j ( p) | j ∈ ,  is an index set}

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that associates each object p with a family of clumps which may be empty, finite or infinite. The subset in each family of clumps is called an elementary neighborhood of p. The collection {N S( p) | p ∈ V } is referred to as the neighborhood system of V . A binary granulation B : V → 2U ; p → B p induces a partition {B −1 (B p ) | p ∈ V } on V called the induced partition of B [4]. The equivalence class B −1 (B p ) is nothing but the center set C p of B p . In other words, all the points that have the same neighborhood form an equivalence class in V . If we take V = U , the induced partition will result in an interesting ‘covering’ [7]. The collection of subset pairs (B p , C p ) satisfies the following results. 1. The collection {C p | p ∈ U } forms a partition of U . 2. The collection {B p | p ∈ U } forms a covering of U . Not every neighborhood system can induce a topological space. If the intersection of two arbitrary neighborhoods N1 ( p) and N2 ( p) is again a neighborhood of p, then we say that the neighborhood system satisfies the intersection condition and such a neighborhood system can induce a topological space. There is another way to induce a topology, considering the neighborhood system as a sub-base.

12.2.2 Neighborhood Systems and Granular Computing Models In the ancient times, Zeno, Archimedes, etc. were familiar intuitively about granulation of the space and time into infinitesimal granules [8, 12]. This intuition can be mathematically formalized by a family of subsets of a set that satisfies the axioms of topology. This family is denoted by TNS( p). The notion of topology can be defined either in the global version or in the local version [8, 12]. A topology τ on a set X is a family of subsets that satisfies the axioms of topology. These subsets are called open sets in τ . This is the global version. Again, a topology can also be defined in terms of a topological neighborhood system (TNS) where each point p is associated with a family of subsets defined as TNS( p) that satisfies the (local version) axioms of topology. First or Local GrC model and Second or Global GrC model were derived from these definitions [8, 12]. In general, granular computing is written as GrC in short. The 3-tuple (V, U, B) where B is a neighborhood system is called Local GrC Model. If V = U , then the 3-tuple is reduced to (U, B). If the neighborhood system satisfies the topological axioms, then it becomes TNS. If B is a family of subsets of U , then the pair (U, B) is the Global GrC Model. B is also called a partial covering. It is clear that the Global GrC Model is a special case of Local GrC Model. Partitions and simplicial complexes are examples of such a model [8, 12]. The Third GrC Model or Binary GrC Model [8, 12] was inspired by Heisenberg’s uncertainty principle which states that the momentum and position of a particle

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cannot both be measured accurately at the same time. A precise measurement of the momentum (or position) at a time can only determine a neighborhood of position (or momenta). If B is a BNS, then the 3-tuple (U, V, B) is called the Third GrC Model. In computer science, binary relations are often represented such as graphs, networks and forests. Thus, this model results in most of the mathematical structures in computer science [2, 3]. Instead of a single binary relation, if we consider B to be a set of binary relations, then the model (U, B) is referred to as Fourth GrC Model or Multi-binary GrC Model [8, 12]. Because of its usefulness in databases, this model is also called Binary Granular Data Model (BGDM) [8, 12]. If the relations are equivalence relations, then it is called Granular Data Model (GDM) [8, 12]. Lin [8, 12] discussed up to Ninth GrC Model based on different mathematical ideas. Among all these models, the category theory-based model (i.e., Eighth GrC Model) was proposed to be the final GrC model.

12.3 Algebraic Quotient Space in Granular Computing There are different granular computing models based on different attributes and structures. One of the most important theoretical models of granular computing is the topological quotient space theory. It is discussed on a topological structure and based on an equivalence relation. With the popularity and advancement of granular computing, it was necessary to define a generic granular system model expressing the common properties. Chen et al. [9] proposed a generic abstract granular system model. We restate their definition below. A granular system can be defined as a 4-tuple  = (U, FR , TR , N ) where U = {x1 , x2 , . . . , xn } is the universe; R is the granulation rule; FR is the granule attribute which is a group of functions { f 1 , f 2 , . . . , f n } belonging to each of the granules on granular layer [U ] R ; TR is the inter-granule structure on layer [U ] R and N is the inter-layer structure. The inter-granule structure TR is the most important and complicated element of a granular system. It is an internal network on [U ] R . In topological quotient space model, it is defined as a topology, whereas in the algebraic quotient space model, it is defined as an algebra. Among all the layers in the structure of a granular system, {U } is the coarsest layer while the partition [U ] R = {{x1 }, {x2 }, . . . , {xn }} is the finest. The rough set model (U, F) and the topological quotient space model (U, F, T ) are both subset models of (U, FR , TR , N ).

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12.3.1 Algebraic Quotient Space Model Algebra has many applications in the field of computer science. In the meantime, algebra is broadly used in granular computing to describe inter-granule structures. Replacing the topology T in the topological quotient space model (U, F, T ) with an algebraic operator ◦, Chen et al. [9] proposed algebraic quotient space model as given below. Given a granular system (U, FR , TR , N ) where ◦ is an algebraic operator and R is a congruence relation, based on the finest granularity (U, F, ◦), we can define (1) the granular layer mapping [U ] R : U → U/R; (2) the granular attribute mapping F → FR ; (3) the inter-granule structure mapping ◦ R : ∀a, b ∈ U, p(a ◦ b) = p(a) ◦ R p(b), where p : U → [U ] R is a natural mapping. Then ([U ] R , FR , ◦ R ) is defined as an algebraic quotient space of (U, F, ◦).

12.3.2 Algebraic Operator-Based Quotient Map The inter-granular structure can be described with the help of the map ◦ R as defined above. Chen et al. [9] redefined homomorphism and congruence relation based on ◦ and ◦ R as defined below. If a map p : U → [U ] R such that for all a, b ∈ U, we have p(a ◦ b) = p(a) ◦ R P(b) where (U, ◦) and ([U ] R , ◦ R ) are two algebras, then we say that they are homomorphic to each other. On the other hand, if we consider the algebra (U, ◦) and R as an equivalence relation on it, and if a Rb → (a ◦ c)R(b ◦ c), (c ◦ a)R(c ◦ b), then R is called a congruence relation on (U, ◦). If the granulation rule is a congruence relation on a granular structure instead of an equivalence relation, then it becomes easier to have consistency between two algebraic operators. To attain the consistency between two algebras (U, ◦) and ([U ] R , ◦ R ), we must be able to define a homomorphism between them. If R is an equivalence relation on the quotient space (U, ◦), then there exists a quotient operator ◦ R of ◦ on [U ] R if and only if R is a congruence relation. From this statement, we can conclude that to obtain a homomorphism we should have a congruence relation R other than an equivalence relation on the space (U, ◦). So, in the algebraic quotient space theory, to obtain the homomorphism between the original space and the quotient space, the relation must be a congruence relation. Also from [9], we can add the following two results regarding congruence relations. (i) Intersection of finite congruence relations is a congruence relation. (ii) Union of finite congruence relations is again a congruence relation if we take the transitive closure on the unions.

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12.4 Interactive Granular Computing and Neighborhood Systems Recently, Skowron et al. [16] and Skowron and Dutta [17] introduced interactive granular computing. They adapted approximation space and rough set theory to study interactive granular computing. Complex granules are introduced in interactive granular computing as extensions of granular computing. Uses of rough sets in [16, 17] are naturally raising questions on the existence of interactive granular computing on topological structures of crisp sets. Moreover, connections of general topology and closure space with relations by Allam et al. [19, 20] extend the possibility of interactive granular computing on crisp sets. Also, algebraic structures in granular computing [21] using relations open the path of granular computing based on topology-based algebraic structures. In the case of graph theoretical granular computing [18], the existence of vertex neighborhood and related ideas connect ample scopes of topological graph theory in interactive granular computing. One may refer to Zhang and Zhang [22] for extensive literature on quotient space-based granular computing. Recently, Skowron et al. [24] modeled rough set-based approximation in granular computing with the help of neighborhood systems. Similarly, El-Bably et al. [23] studied granular computing and approximation spaces with the help of closure operators of topology. In the meanwhile, we observe a few scopes of theoretical development in granular computing, which will be proposed in the next section. Thus, topologists have many scopes to develop new theories using crisp sets in interactive granular computing.

12.5 Some Open Questions Inspired by Microscopy, Biology and Neuroscience Recently, Sun and Xe [25] used granular computing-based method in gene selection. Rizzi et al. [26] and Martino et al. [27] also studied some biological problems from the perspective of granular computing. In 2011, Yao [28] went a step further and connected granular computing with artificial intelligence. Thus, it can be concluded that several biological phenomena may be explained using granular computing via artificial intelligence in near future. But two research articles [29, 30] attracted our attention from the perspective of granular computing, and they indirectly raised some questions on existing theories of granular computing from the viewpoint of biology. In [30], Hell and his collaborators designed a technique named stimulated emission depletion (STED) microscopy. In this technique, they used two beams of light, one to stimulate fluorescent molecules, and another to immediately turn most of them off in such a way that only those at the center of the light beam continued to shine [30, 31]. Because of the pioneering research of Hell and his collaborators [30], the first super resolution microscope only exceeded the diffraction limit by around a factor of two in 2000 [31]. Later, Betzig et al. [29] introduced a method for optically imaging

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intracellular proteins at nanometer spatial resolution. In this method, they activated numerous sparse subsets of photoactivatable fluorescent protein molecules (localized to 2–25 nm), and then bleached. Later, they aggregated position information from all subsets. This information was then assembled which resulted in a super resolution image. These two techniques were very important from the perspective of microscopy because they broke down the barrier of Abbe’s limit for microscopy and as a result, we have now microscopes of higher resolutions than those of earlier days. Due to these pioneering contributions, Eric Betzig, Stefan W. Hell and William E. Moerner were awarded the Nobel Prize in Chemistry 2014 having the citation as follows [32]: “for the development of super-resolved fluorescence microscopy.” Similar to microscopy, another research area of neuroscience named ‘Optogenetics’ has started to use granulation techniques [33] to study various kinds of brain disorders. Moreover, the connection between adaptive learning and granular computing in [34] yields the useful future of granular computing in machine learning. In the case of adaptive learning (basically perception), spatial granulation [35] is highly useful. Thus, it is the need of the hour to develop some new theories in granular computing which may become more feasible to the natural world than that of theoretical world. But we don’t deny the importance of existing theories of granular computing. It must be remembered that Carnot heat engine [36], a theoretical model, was the significant contribution behind heat engines which are available to us. Thus, existing theories must be reconstructed or new theories may be proposed to serve broader purposes in nature. Thus, we propose the following open questions in granular computing from the perspective of crisp set and crisp set-based topology. (i) How to choose a granular attribute FR which is a group of functions { f 1 , f 2 , . . . , f n } belonging to each of the granules of granular layer [U ] R in [9]? (ii) How to indicate inter-granule structure TR [9] on layer [U ] R ? (iii) How to establish inter-layer structure N in [9] where each layer has distinct granulation structure? (iv) How to develop granular attribute mapping F → FR in case of adaptive learning of artificial intelligence with reference to [9, 34]? (v) What are the inter-granule structures in [9] if they are distinct? (vi) Is there any biological impossibility while inserting inter-granule structures in the case of distinct granules? (vii) How to connect theories of granular computing with adaptive learning of artificial intelligence? (viii) What is the perspective of granular computing when neighborhood systems of any two distinct elements are not distinct? (ix) What are the crisp set-based topological methodologies in granular computing in the presence of attributes? The above questions may help the experts of granular computing to develop more realistic ideas using crisp sets. There is no specific granulation process available in granular computing. Experts use various methods for granulation. We refer to Liu et al. [37], Chen et al. [38] and Yao [39]. In [37], the distance measure between two granules is considered as

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a classification technique for granular computing. Chen et al. [38] suggested a classification method based on algebraic granule structure for granular computing. Also in [39], Yao wrote that multiple levels of granules with varied granularities make up a granular structure. Numerous granular structures result in multi views, but a single granular structure offers a multilayer knowledge and depiction of the reality or a situation. From this, we can conclude that there is no specific granulation process in granular computing. Thus, topologists have ample scopes to develop various granulation techniques using generalized ideas of topology and allied areas.

12.6 Conclusion In this survey, we discussed the granular system model from the crisp topological viewpoint with the help of the universe, granulation rule, attributes of granular layer, inter-granular structure layer and the inter-layer structure. Then, some definitions of algebraic quotient mapping regarding the granular structure as well as some results regarding equivalence relations and congruence relations are discussed. Moreover, we tried to establish possible connections between interactive granular computing and crisp set-based topology using relations. At present, granular computing models have some attractive ideas regarding many areas of science. Out of all the models, quotient space theory and its generalized structures connect topology with granular computing. So, we must take care of the topological aspects of granular computing to have a wonderful blossom of it from a multidisciplinary point of view. At the end, we hope that our survey on crisp set-based topological structures in granular computing will attract the attention of topologists.

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Chapter 13

On Topological Index of Naturally Occurring Zeolite Material [4, n] Saitanya K. Bharadwaj and Santanu Acharjee

Abstract A topological index is a numeric quantity which characterizes the whole structure of a chemical graph. Here, we compute Randi´c index and its generalized forms, first Zagreb index, augmented Zagreb index, harmonic index, hyper Zagreb index, atom-bond connectivity index, geometric-arithmetic index, forgotten index of zeolite material [4, 2] and zeolite material [4, n], where n ≥ 3. We also calculate the fourth version of the atom-bond connectivity index and the fifth version of the geometric-arithmetic index of zeolite material [4, n], where n ≥ 5. Keywords Topological index · Chemical graph · Zeolite

13.1 Introduction Zeolites are hydrated crystalline aluminosilicates with pores having the general formula Mx/n [(Al O2 )x (Si O2 ) y ].m H2 O, where M is any metal, Al is aluminium and Si is silicon. Structurally, Al O4 and Si O4 units are interlinked by sharing oxygen atoms giving a three-dimensional inorganic polymer. In particular, zeolites have relatively open, three-dimensional crystal structures built from the elements aluminium, oxygen and silicon with alkali or alkaline-earth metals (such as sodium, potassium and magnesium). The sharing of the tetrahedral alumina and silica resulted in orderly distributed micropores with 0.3–0.2 nm diameter and voids which are occupied by alkali cations and water molecules. Depending on the Si/Al ratio, different types of zeolites are formed and their structures vary [1, 2] (Fig. 13.1). Zeolite can be synthesized as well as obtained naturally. Zeolite’s formation may occur under various conditions such as hydrothermal alteration of basalts, metamorS. K. Bharadwaj Department of Chemistry, Pragjyotish College, Guwahati 781009, Assam, India e-mail: [email protected] S. Acharjee (B) Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_13

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phism of lava pile, deep-sea sediments, low-temperature open hydrologic systems and burial diagenesis [3]. Most natural zeolites are formed as a result of volcanic activity. Volcanic eruption breaks the magna rocks which transform into solid crystalline zeolites over thousands of years. Naturally occurring zeolite materials occupy a large volume of the earth’s crust, known as clay minerals. They are classified on the basis of their structural varieties, cationic species, etc. [4].

Fig. 13.1 a Silica and alumina tetrahedra, b sharing of oxygen of these two tetrahedra and c two-dimensional representation of the arrangement of alumina and silica tetrahedra

As far as the structure of zeolite is concerned, it has two basic building units viz. Primary Building Unit (PBU) and Secondary Building Unit (SBU). The PBUs are the (Si O4 )4− and (Al O4 )5− tetrahedra. These two tetrahedra combine by sharing oxygen to form a simple geometric arrangement which is known as the SBU. Twentythree different SBUs are known to date [5]. These are just like tetragon, hexagon, etc. (Fig. 13.2). Apart from PBU and SBU, some zeolites also contain other building units, i.e. Composite Building Units (CBU) which are formed by SBUs. Based on the SBU and CBU, different types of zeolites are formed or naturally found with dissimilar morphology with different pore sizes [2]. The first application of zeolites was done in 1954 as adsorbents for industrial separations and purifications. Since then, they have occupied a large volume in material science as well as in catalysis due to their structural varieties. In addition to the better adsorbing capacity, they are also found to be ion exchangers and molecular sieves.

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Fig. 13.2 Different orientations of Secondary Building Blocks. The intersections of lines represent the centre of tetrahedral unit a tetragon, b hexagon, c octagon, d cuboid and e 8–8 cuboid

Zeolites have been used for hydrocarbon separation [6], air separation [7] and water purification [8]. Zeolites found their space in medicine and have been used for cancer treatment [9]. Zeolite-based commercial products Hemosorb and QuikClot are applied to wounds (in accidents or surgeries) and are said to cause an instant cessation of bleeding. Zeolites form solid acid when they are exchanged with protons (H+) and are used extensively in the petrochemical industry for cracking hydrocarbon fractions into fuels and in the preparation of raw chemical feedstock for various manufacturing processes. Zeolites are also used to capture carbon dioxide and convert into important products [10]. One may refer to [11] for an extensive review of zeolite materials. Recently, Ates et al. [12] investigated the role of natural zeolite material in the adsorption of arsenic from an aqueous solution. The variation in the composition and structure of zeolite leads to the development of many selective and specific catalysts. The composite atoms or molecules arrange or attach in such a way that they form a channel or pore having a specific dimension. Weisz and Frilette termed this structure-activity relation of zeolites as Shape-Selective Catalysis. Shape selectivity is dependent on the pore size; the decrease in pore size increases the selectivity [13]. One of the best examples of zeolite is the alkylation of toluene to give para-xylene selectively. The high selectivity is due to the presence of channels or pores inside the zeolite material. Naturally occurring zeolites are known as clay minerals. These are the result of weathering the rocks. As mentioned earlier, structural units of clay are Si O4 tetrahedra and Al O4 octahedral, which form sheet or layer structures like zeolites. Depending on the layering clay materials differ in structure and properties. The layering is shown in Fig. 13.3. There is interlayer spacing where water and other small ions can be absorbed by the clay minerals. As shown in Fig. 13.3, 2:1 layering gives smectite-type clay and 1:1 layering results in kaolinite-type clay. The most common smectite clay is montmorillonite. The arrangement of tetrahedral and octahedral units can be drawn in two-dimensional papers specifying the atoms as shown in Fig. 13.4. Out of all clay minerals, montmorillonite can expand maximum by absorbing water molecules in the interlayer spacing. It has been found that montmorillonite has a very high specific surface, cation exchange capacity and affinity to water. Therefore, it has been well-explored in terms of physical and chemical properties.

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Fig. 13.3 a Tetrahedral silica unit, b arrangement of tetrahedral sheet, c octahedral alumina unit, d arrangement of octahedral unit and e, f two layering possibilities of tetrahedral and octahedral sheets to give structurally diverse clay materials

Fig. 13.4 Three-dimensional structure of montmorillonite clay

Apart from catalysis, clay minerals exhibit many properties such as ion exchange, intercalation, swelling, cohesion, flocculation and dispersion. Clay minerals also show both Brönsted and Lewis acidity. The Lewis acidity is due to the presence of A13+ and Fe3+ at the crystal edges, and the Brönsted acid character of clays arises mainly due to the dissociation of the intercalated water molecules coordinate to cations. Modifying the cationic site, the acidity of the clay minerals has been varied and applied in various chemical reactions with great efficiency [14, 15]. Therefore, theoretical calculations delineating the structure-property relationship are found to be much more important for clay minerals. The fundamental concept of structure-property relationship in chemistry was first shown by Crum Brown

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and Fraser [16]. Quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs) are mathematical correlations between molecular topology and biological activity or physicochemical properties [17]. Chemical graph theory is a branch of mathematics which deals with graph theory from the perspective of a chemical phenomenon, and topological indices is one of the subtopics of chemical graph theory from the perspective of various topological properties of molecules [18]. Chemical graph theory is found to be very important in theoretical chemistry as it provides an important tool, i.e. topological indices. A topological index can be defined as a real-valued function f : G → R+ , which maps every molecular structure G to a non-negative real number [19]. These indices are of much importance for chemical significance. The indices are found to correlate different physical and chemical properties of a material with its structural arrangement. Topological indices are calculated for nanostructured polynuclear hydrocarbons [20], triglycerides [21], graphene [22], fullerene [23, 24] and bicyclic compounds [25], to mention a few. Recently, a reduced hyper-Zagreb index has been calculated for silicate network [26]. These reported calculated indices are expected to correlate the molecular structure and hence physical and chemical properties of silicate materials. Very recently, Saeed et al. [27] calculated various Zagreb indices for Z S M-5, a synthetic zeolite, and correlated the indices with the chemical structure of Z S M-5. Another report [28] revealed the calculation of various Zagreb indices for the planar octahedron network, triangular prism network and hex planar octahedron network. It is expected that the computed data will be helpful for people working in computer science and chemistry who encounter hex-derived networks [29]. A similar calculation for m th chain hex-derived network of the third type is also reported recently [29]. Different topological indices for H -naphthalenic nanotube and H A(C5c6c7) nanotube are calculated which are useful for physical features, biological activities and chemical reactivates of the substance [30, 31].

13.2 Preliminaries To apply topology and graph theory, the three-dimensional layered structure of zeolite materials of Fig. 13.4 are drawn in two-dimension as shown in Fig. 13.5. Silicon and aluminium are present at the centre of the tetrahedra and have no effect on the shape and size of the zeolite material, thus, they are left out in the transformation from three dimension to two dimension. Moreover, selectivity and other related properties will remain the same as the zeolite-montmorillonite clay. Randi´c [32] proposed “branching index” in the year 1975. It is a topological index R (also R−1 and R −12 ), and it is used to measure the branching of the carbon-atom skeleton of saturated hydrocarbons. Later, Ballobás and Erd˙os [33] generalized this index c index”. General Randi´c index [33] is given by Rα (G) =  as “general Randi´ α (d d ) . u v uv∈E(G)

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Fig. 13.5 Two-dimensional view of layered structure of clay minerals

If α = −1 , then it reduces to Randi´c index R −12 . The first Zagreb index M1 and the 2 second Zagreb index M2 were introduced by Balaban [34]. If α = −1, then general Randi´c indexreduces to second Zagreb index M2 . The first Zagreb index is defined as M1 (G)= uv∈E(G) (du + dv ). The atom-bond connectivity index (ABC)  is a degree-based topological index 

v −2 [35], and it is denoted by ABC(G)= uv∈E(G) du +d . The atom-bond connectivity du dv index plays a crucial role in the stability of branched and linear alkenes [36]. Recently, Vuki˘cevi´c and Furtula [37] introduced “geometric-arithmetic index”. The geometric√  du dv arithmetic index of a graph G is given by G A(G) = uv∈E(G) 2 du +dv . Zhong [38] defined

harmonic index of a graph G as H (G) =



uv∈E(G) 

[39] of a graph G is defined as AZ I (G) = 

2 ( du +d ). The augmented Zagreb index v

uv∈E(G)

du dv ( du +d )3 . The hyper-Zagreb index v −2

is given by H M(G) = uv∈E(G) (du + dv )2 , and it was introduced in [40]. Frutula and Gutman [41]  introduced the forgotten topological index F in 2015. They defined it as F(G) = uv∈E(G) {(du )2 + (dv )2 }. 

If NG (u)= {v ∈ V (G) : uv ∈ E(G)}, then we denote Su = v∈N (u) dv . The fourth G version of the atom-bond connectivity index( ABC4 ) introduced by Ghorbani et al. 

[42] and it is defined as ABC4 (G)= uv∈E(G)

Su +Sv −2 . Su Sv

In 2011, Graovac et al. [43]

defined the fifth version of topological index G A5 as G A5 (G)=



√

u Sv 2 SuS+S . uv∈E(G) v

13.3 Main Results We denote a sheet of zeolite material with four rows and n columns as zeolite material [4, n]. A zeolite material [4, n] is shown in Fig. 13.6. Here, a row and a column are indicated as a horizontal arrangement of atoms and a hexagonal structure, respectively. If G is any graph, then V (G) and E(G) are vertex set and edge set of G, respectively. If G indicates zeolite material [4, n], where n ≥3, then |V (G)| = 6n + 2 and |E(G)| = 13n + 2(n − 1) + 1 = 15n − 1. Let du be the degree of the vertex u of the graph G (Table 13.1).

13 On Topological Index of Naturally Occurring Zeolite Material [4, n]

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Fig. 13.6 Two-dimensional arrangement of zeolite material [4, n] Table 13.1 Vertex-degree table of zeolite material [4, 2] du 3 4 Number of vertices

4

7

8

2

Table 13.2 Edge-degree table of zeolite material [4, 2] (du , dv ) (3, 3) (3, 4) (3, 7) (4, 4) Number of edges

2

4

4

10

(7, 4)

(7, 7)

8

1

Table 13.3 Vertex-degree table of zeolite material [4, n], where n ≥ 3 du 3 4 Number of vertices

4

7 2n − 2

4n

To make the calculations easier, now we have the following tables (Table 13.2). (i) For zeolite material [4, 2] (ii) For zeolite material [4, n], where n ≥ 3 (Tables 13.3 and 13.4). Now, we have the following theorems. Theorem 1 If G indicates zeolite material [4, n], where n ≥ 3, then the general Randi´c index Rα of G is given by Rα (G) = 8(12)α + 4(21)α + (2n + 4)(16)α + (12n − 16)(28)α + (n − 1)(49)α . Proof By using definition of general Randi´c index, we have

Table 13.4 Edge-degree table of zeolite material [4, n], where n ≥ 3 (du , dv ) (3, 4) (3, 7) (4, 4) (7, 4) Number of edges

8

4

2n + 4

12n − 16

(7, 7) n−1

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Fig. 13.7 Two-dimensional graphs of R 1 (G), R−1 (G) and R −1 (G) 2

Rα (G) =

 uv∈E(G)

2

(du dv )α .

= 8(12)α + 4(21)α + (2n + 4)(16)α + (12n − 16)(28)α + (n − 1)(49)α .



Corollary 1 If G indicates zeolite material [4, n], where n ≥ 3, then the Randi´c index R1 (G) = 417n − 253. Proof By putting α = 1 in Theorem 1, we have the proof.



Corollary 2 If G indicates zeolite material [4, n], where n ≥ 3, then R 21 (G) = √ √ √ √ (15 + 24 7)n + (16 3 + 4 21 − 32 7 + 9). Proof By putting α =

1 2

in Theorem 1, we have the proof.



Corollary 3 If G indicates zeolite material [4, n], where n ≥ 3, then R−1 (G) = 225n + 101 . 392 196 Proof By putting α = −1 in Theorem 1, we have the proof.



Corollary 4 If G indicates zeolite material [4, n], where n ≥ 3, then R −12 (G) =     9 6 √6 √4 − √8 + √4 n + . + + 14 7 7 3 7 21 Proof By putting α =

−1 2

in Theorem 1, we have the proof.



Two and three-dimensional graphs of R 21 (G), R−1 (G) and R −12 (G) are shown in Figs. 13.7 and 13.8. Theorem 2 If G indicates zeolite material [4, n], where n ≥ 3, then the first Zagreb index of G is M1 (G) = 162n − 62.

13 On Topological Index of Naturally Occurring Zeolite Material [4, n]

237

Fig. 13.8 Three-dimensional graphs of R 1 (G), R−1 (G) and R −1 (G) 2

2

Proof By definition of first Zagreb index, 

M1 (G)= uv∈E(G) (du + dv ) = 8(7) + 4(10) + (2n + 4)(8) + (12n − 16)(11) + (n − 1)(14) = 162n − 62.



Theorem 3 If G indicates zeolite material [4, n], where n ≥ 3, then the augmented − 3395253037 . Zagreb index of G is AZ I (G) = 7268377n 15552 11664000 Proof By definition of the augmented Zagreb index, AZ I (G) = =8 =

 12 3



 uv∈E(G)

5

+4

7268377n 15552

−

du dv du +dv −2

 21 3 8

3

+ (2n + 4)

 16 3 6

+ (12n − 16)

3395253037 . 11664000

 28 3 9

+ (n − 1)

 49 3 12



Theorem 4 If G indicates zeolite material [4, n], where n ≥ 3, then the harmonic + 398 . index of G is H (G) = 435n 154 385 Proof By definition of the harmonic index,    2 H (G)= uv∈E(G) du +d v

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=8 =

2 7

435n 154

+4

+

2 10

+ (2n + 4)

2 8

+ (12n − 16)

2 11

+ (n − 1)

2 14



398 . 385

Theorem 5 If G indicates zeolite material [4, n], where n ≥ 3, then the hyperZagreb index of G is H M(G) = 1776n − 1084. Proof By definition of the hyper-Zagreb index, 

H M(G) =

uv∈E(G)

(du + dv )2

= 8(7)2 + 4(10)2 + (2n + 4)(8)2 + (12n − 16)(11)2 + (n − 1)(14)2 = 1776n − 1084.



Theorem 6 If G indicates zeolite material [4, n], where atom-bond then the   n≥  3, √ √ 3 2 3 18 2 √ connectivity of G is ABC(G)= 2 + 7 + 7 n + 8 21 + 4 53 − 2 7 3 +  √ 6 − √247 . Proof By definition of the atom-bond connectivity index,   v −2 ABC(G)= uv∈E(G) du +d du dv      5 8 6 9 = 8 12 + 4 21 + (2n + 4) 16 + (12n − 16) 28 + (n − 1) 12 49       √ √ √ 3 2 = + 2 7 3 + √187 n + 8 21 + 4 35 − 2 7 3 + 6 − √247 . 2



Theorem 7 If G indicates zeolite material [4, n], where√n ≥ 3, then the√geometric√ √ arithmetic index of G is G A(G)=(3 + 4811 7 )n + (3 + 327 3 − 6411 7 + 4 521 ). Proof By definition of the geometric-arithmetic index, √



G A(G) =

uv∈E(G)

√

u dv 2 dud+d v

√

√

√

√

= 8 × 2 712 + 4 × 2 1021 + (2n + 4) × 2 816 + (12n − 16) × 2 1128 + (n − 1) × 2 1449

 = 3+

√  48 7 n 11

 + 3+

√ 32 3 7

−

√ 64 7 11

+

√  4 21 . 5



Theorem 8 If G indicates zeolite material [4, n], where n ≥ 3, then the forgotten index of G is F(G) = 942n − 578. Proof By definition of the forgotten index, 

F(G) =

uv∈E(G)

{(du )2 + (dv )2 }.

13 On Topological Index of Naturally Occurring Zeolite Material [4, n]

239

Fig. 13.9 Two-dimensional graphs of Theorems 2–8

Fig. 13.10 Three-dimensional graphs of Theorems 2–8

= 8(32 + 42 ) + 4(32 + 72 ) + (2n + 4)(42 + 42 ) + (12n − 16)(42 + 72 ) + (n − 1) (72 + 72 )

= 942n − 578.



Two and three-dimensional graphs of Theorems 2–8 are shown in Figs. 13.9 and 13.10. Theorem 9 If K indicates zeolite material [4, 2], then (i) Rα (K ) = 2(9)α + 4(12)α + 4(21)α + 10(16)α + 8(28)α + (49)α ,

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(ii) R1 (K )= 583, √ √ √ (iii) R 21 (K ) = 53 + 8 3 + 16 7 + 4 21, (iv) R−1 (K ) = (v) R −12 (K ) =

5917 , 3528 139 42

+

√2 3

+

√4 7

+

√4 , 21

(vi) M1 (K ) = 262, (vii) AZ I (K ) = (viii) H (K ) =

7570428419 , 11664000

15493 , 2310

(ix) H M(K ) = 2472, (x) ABC(K ) =

4 3

   2 + 8 11 + 5 23 + 2 53 +

(xi) G A(K ) = 13 +

√ 16 3 7

+

√ 32 7 11

+

√ 2 3 7

+

12 √ , 7

√ 4 21 , 5

(xii) F(K ) = 1306.

13.4 Neighbourhood-Based Topological Index In this section, we calculate ABC4 (H ) and G A5 (H ). Here, H indicates zeolite material [4, n], where n ≥ 5. Thus, we have the following table. Summation degrees of edge endpoints Number of edges

(15, 15) 6

(21, 31) 4

(22, 31) 4(n–4)

(22, 22) 2(n–5)

(22, 30) 8

(15, 21) 4

(15, 22) 8

(28, 30) 8

(15, 30) 4

(28, 31) 8(n–3)

(21, 22) 4

(30, 30) 2

(21, 30) 4

(31, 31) (n–3)

Theorem 10 If H indicates zeolite material [4, n], where n ≥ 5, then the fourth version of the atom-bond connectivity index of H is       √ √ 57 2 15 2 57 ABC4 (H ) = 4 217 + 2 102 + + 1142 n + 20 651 − 12 217 − 341 31 √ 34 √ 14     √ √ 4 2 82 8 102 + 4 10 + 2 231 + 4 14 + 3 35 + 3 5 + 4 5 7 + √815 − 6 3115 − 341 33 33 √ √ √  5 42 + 1558 + 2 1586 . 11

13 On Topological Index of Naturally Occurring Zeolite Material [4, n]

241

Proof By definition of the fourth of atom-bond connectivity index,   v −2 ABC4 (H )= uv∈E(G) Su +S Su Sv        28 34 35 43 41 49 50 = 6 225 + 4 315 + 8 330 + 4 450 + 4 462 + 4 630 + 4 651 + 2(n − 5)        42 50 51 56 57 58 60 +8 660 +4(n − 4) 682 + 8 840 + 8(n − 3) 868 + 2 900 + (n − 3) 961 . 484        √ √  57 102 2 15 42 2 57 102 = 4 217 + 2 341 + 31 + 11 n + 20 651 − 12 217 − 8 341 + 4 √ √    √ √ √ √ 4 34 2 14 10 82 14 4 7 35 5 √8 − 6 15 − 5 42 + 58 + + 2 + 4 + + + + 33 231 33 3 3 5 31 11 15 15 √  2 86 .  15 Theorem 11 If H indicates zeolite material [4, n], where n ≥ 5, the fifth version of topological index   √  of H√ is √ √ √ √ √ G A5 (H )= 3 + 32 59217 + 8 53682 n + 8 3 2 + 2 335 + 8 1770 + 8 13165 + 16 29210 −  √ √ √ √ √ 96 217 + 16 37330 + 8 43462 + 2 13651 − 32 53682 − 5 . 59 Proof By definition of the fifth version of topological index, G A5 (G)=

√

u Sv 2 SuS+S . v

√ √ √ √ 315 330 450 630 462 + 8 × 2 37 + 4 × 2 45 + 4 × 2 43 + 4 × 2 51 36 √ √ √ √ √ 651 484 660 682 840 + 4 × 2 52 + 2(n − 5) × 2 44 + 8 × 2 52 + 4(n − 4) × 2 53 + 8 × 2 58 √ √ √ 961 868 900 + 8(n − 3) × 2 59 + 2 × 2 60 + (n − 3) × 2 62 .   √  √ √ √ √ √ √ √ 32 217 8 682 8 2 2 35 8 70 = 3 + 59 + 53 n + 3 + 3 + 17 + 8 13165 + 16 29210 − 96 59217 +  √ √ √ √ 16 330 + 8 43462 + 2 13651 − 32 53682 − 5 .  37

=6×2

√

 uv∈E(G)

225 30

+4×2

√

Two and three-dimensional graphs of ABC4 (H ) and G A5 (H ) are given below in Figs. 13.11 and 13.12.

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Fig. 13.11 Two-dimensional graphs of ABC4 (H ) and G A5 (H )

Fig. 13.12 Three-dimensional graphs of ABC4 (H ) and G A5 (H )

13.5 Conclusion In this chapter, a brief idea of the molecular composition and structure of zeolite is presented. The importance of zeolite is drawn by citing different applications. Clays are naturally occurring zeolites having layering structures. The layering structure of montmorillonite clay provides extra stability and tunable chemical reactivities. We calculated Randi´c index and its generalized forms, first Zagreb index, augmented Zagreb index, harmonic index, hyper-Zagreb index, atom-bond connectivity index, geometric-arithmetic index, forgotten index of naturally occurring zeolite material [4, 2] and zeolite material [4, n], where n ≥ 3. We also calculated the fourth version of the atom-bond connectivity index and the fifth version of the geometric-arithmetic index of zeolite material [4, n], where n ≥ 5. Since this study calculated the topo-

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logical indices of zeolite materials for the first time, we hope it will help theoretical chemists for QSAR and QSPR study. Thus, a more deep study will be needed in this direction on various aspects of zeolite material.

References 1. Flanigen, E.M.: Molecular sieve zeolite technology-the first twenty-five years. Pure Appl. Chem. 52(9), 2191–2211 (1980) 2. Moshoeshoe, M., Nadiye-Tabbiruka, M.S., Obuseng, V.: A review of the chemistry, structure, properties and applications of zeolites. Am. J. Mater. Sci. 7(5), 196–221 (2017) 3. Hay, R.L.: Geologic occurrence of zeolites and some associated minerals. In: Studies in Surface Science and Catalysis, vol. 28, pp. 35–40. Elsevier (1986) 4. Weckhuysen, B.M., Yu, J.: Recent advances in zeolite chemistry and catalysis. Chem. Soc. Rev. 44(20), 7022–7024 (2015) 5. Baerlocher, C., McCusker, L.B., Olson, D.H.: Atlas of Zeolite Framework Types. Elsevier (2007) 6. Dixon, J.B., Schulze, D.G.: Soil Mineralogy with Environmental Applications. Soil Science Society of America Inc. (2002) 7. Ribeiro, L.E.B., Alcantara, G.P.D., Andrade, C.M.G., Fruett, F.: Analysis of the planar electrode morphology applied to zeolite based chemical sensors. Sens. Transducers. 193, 80–85 (2015) 8. Visa, M.: Synthesis and characterization of new zeolite materials obtained from fly ash for heavy metals removal in advanced wastewater treatment. Powder Technol. 294, 338–347 (2016) 9. Paveli´c, K., Hadžija, M., Bedrica, L., Paveli´c, J., Diki´c, I., Kati´c, M., Kralj, M., Bosnar, M.H., Sanja, K., Poljak-Blaži, M., Križanac, S., Stojkovic, R., Jurin, M., Subotic, B., Coli´c, M.: Natural zeolite clinoptilolite: new adjuvant in anticancer therapy. J. Mol. Med. 78(12), 708– 720 (2001) 10. Pera-Titus, M.: Porous inorganic membranes for CO2 capture: present and prospects. Chem. Rev. 114(2), 1413–1492 (2014) 11. Danilczuk, M., Długopolska, K., Ruman, T., Pogocki, D.: Molecular sieves in medicine. Mini. Rev. Med. Chem. 8(13), 1407–1417 (2008) 12. Ates, A., Özkan, I., Canbaz, G.T.: Role of modification of natural zeolite in removal of arsenic from aqueous solutions. Acta Chim. Slov. 65(3), 586–598 (2018) 13. Weisz, P.B., Frilette, V.J.: Intracrystalline and molecular-shape-selective catalysis by zeolite salts. J. Phys. Chem. 64(3), 382 (1960) 14. Kumar, B.S., Dhakshinamoorthy, A., Pitchumani, K.: K10 montmorillonite clays as environmentally benign catalysts for organic reactions. Catal. Sci. Technol. 4(8), 2378–2396 (2014) 15. Balogh, M., Laszlo, P.: Organic Chemistry Using Clays. Springer, Berlin (1993) 16. Brown, A.C., Fraser, T.R.: On the connection between chemical constitution and physiological action; with special reference to the physiological action of the salts of the ammonium bases derived from strychnia, brucia, thebaia, codeia, morphia, and nicotia. J. Anat. Physiol. 2(2), 224–242 (1868) 17. Dearden, J.C.: The use of Topological Indices in QSAR and QSPR Modeling. In: Advances in QSAR Modeling, pp. 57–88. Springer, Cham (2017) 18. Hayat, S., Imran, M.: On degree based topological indices of certain nanotubes. J. Comput. Theor. Nanosci. 12(8), 1599–1605 (2015) 19. Gao, W., Siddiqui, M.K., Imran, M., Jamil, M.K., Farahani, M.R.: Forgotten topological index of chemical structure in drugs. Saudi Pharm. J. 24(3), 258–264 (2016) 20. Soleimani, N., Mohseni, E., Rezaei, F., Khati, F.: Some formulas for the polynomials and topological indices of nanostructures. Acta Chem. Iasi 24(2), 122–138 (2016) 21. Kanna, M.R., Mamta, D., Indumathi, R.S.: Computation of topological indices of triglyceride. Glob. J. Pure Appl. Math. 13(6), 1631–1638 (2017)

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22. Jagadeesh, R., Kanna, M.R., Indumathi, R.S.: Some results on topological indices of graphene. Nanomater. Nanotechnol. 6, 1–6 (2016) 23. Andova, V., Kardoš, F., Škrekovski, R.: Mathematical aspects of fullerenes. Ars Math. Contemp. 11, 353–379 (2016) 24. Baˇc, M., Horvathova, J., Mokrišova, M., Suhanyiova, A.: On topological indices of fullerenes. Appl. Math. Comput. 251, 154–161 (2015) 25. Ma, J., Shi, Y., Wang, Z., Yue, J.: On wiener polarity index of bicyclic networks. Sci. Rep. 6(1), 1–7 (2016) 26. Kulli, V.R.: Reduced second hyper-Zagreb index and its polynomial of certain silicate networks. J. Math. Infor. 14, 11–16 (2018) 27. Saeed, N., Long, K., Islam, T.U., Mufti, Z.S., Abbas, A.: Topological study of zeolite socony mobil-5 via degree-based topological indices. J .Chem. Article ID 5522800 (2021) 28. Dustigeer, G., Ali, H., Khan, M.I., Chu, Y-M.: On multiplicative degree based topologicalindices for planar octahedron networks. Main Group Met. Chem. 43(1), 219–228 (2020) 29. Huo, Y., Ali, H., Binyamin, M.A., Asghar, S.S., Babar, U., Liu, J-B.: On topological indices of mth chain hex-derived network of third type. Front. Phys. 8, 593275 (2020). https://doi.org/ 10.3389/fphy.2020.593275 30. Afzal, F., Alsinai, A., Zeeshan, M., Afzal, D., Chaudhry, F., Cancan, M.: Some new degree based topological indices of h-naphtalenic graph via M-polynomial approach. Eurasian Chem. Commun. 3(11), 800–805 (2021) 31. Parvathi, N., Krishnan, V.L.: Computation of topological indices of HA (C5c6c7) nanotube. Annal. Romanian Soc. Cell Bio. 25(6), 3080–3085 (2021) 32. Randic, M.: Characterization of molecular branching. J. Am. Chem. Soc. 97(23), 6609–6615 (1975) 33. Bollobás, B., Erdos, P.: Graphs of extremal weights. Ars Comb. 50, 225–233 (1998) 34. Balaban, A.T.: Chemical graphs. Theor. Chim. Acta. 53(4), 355–375 (1979) 35. Estrada, E., Torres, L., Rodriguez, L., Gutman, I.: An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. 37A, 849–855 (1998) 36. Estrada, E.: Generalization of topological indices. Chem. Phys. Lett. 336(3–4), 248–252 (2001) 37. Vukiˇcevi´c, D., Furtula, B.: Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 46(4), 1369–1376 (2009) 38. Zhong, L.: The harmonic index for graphs. Appl. Math. Lett. 25(3), 561–566 (2012) 39. Furtula, B., Graovac, A., Vukiˇcevi´c, D.: Augmented Zagreb index. J. Math. Chem. 48(2), 370–380 (2010) 40. Shirdel, G.H., Rezapour, H., Sayadi, A.M.: The hyper-Zagreb index of graph operations. Iran. J. Math. Chem. 4, 213–220 (2013) 41. Furtula, B., Gutman, I.: A forgotten topological index. J. Math. Chem. 53(4), 1184–1190 (2015) 42. Ghorbani, M., Hosseinzadeh, M.A.: Computing ABC4 index of nanostar dendrimers. Optoelectron. Adv. Mater.-Rapid Commun. 4, 1419–1422 (2010) 43. Graovac, A., Ghorbani, M., Hosseinzadeh, M.A.: Computing fifth geometric-arithmetic index for nanostar dendrimers. J. Discrete Math. Appl. 1(1–2), 33–42 (2011)

Chapter 14

q-Rung Orthopair Fuzzy Points and Applications to q-Rung Orthopair Fuzzy Topological Spaces and Pattern Recognition Ezgi Türkarslan, Mehmet Ünver, Murat Olgun, and Seyhmus ¸ Yardımcı Abstract In this chapter, we introduce the concept of q-rung orthopair fuzzy point and propose a Dice similarity measure and a distance measure between q-rung fuzzy sets by using the concept of Choquet integral which is a non-linear continuous aggregation operator. Then, we give some applications on pattern recognition by using q-rung orthopair fuzzy points and the Dice similarity measure. Moreover, we introduce the concept of continuity of a function defined between two q-rung orthopair fuzzy topological spaces at a q-rung orthopair fuzzy point and define the concept of convergence of nets of q-rung orthopair fuzzy points in a q-rung orthopair fuzzy topological space. Finally, we study the relationship between continuity of functions and convergence of nets. Keywords q-rung orthopair fuzzy point · q-rung orthopair fuzzy topology · Dice similarity measure · Pattern recognition

14.1 Introduction Fuzzy set theory [41] that was introduced via a membership function by Zadeh in 1965 is very successful in handling uncertainties or partial belongingness of an element to a set. Then, this theory was extended to intuitionistic fuzzy set E. Türkarslan Faculty of Arts and Science, Department of Mathematics, TED University, 06420 Ankara, Turkey e-mail: [email protected]; [email protected] E. Türkarslan · M. Ünver (B) · M. Olgun · S. ¸ Yardımcı Faculty of Science, Department of Mathematics, Ankara University, 06100 Ankara, Turkey e-mail: [email protected] M. Olgun e-mail: [email protected] S. ¸ Yardımcı e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Acharjee (ed.), Advances in Topology and Their Interdisciplinary Applications, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-99-0151-7_14

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(IFS) theory by Atanassov [1] via a membership function μ A : X → [0, 1] and a non-membership function ν A : X → [0, 1] such that 0 ≤ μ A (x) + ν A (x) ≤ 1 for each x ∈ X . However, the concept of IFS is not sufficient in some cases. Therefore, Yager [39] introduced the concept of Pythagorean fuzzy set (PFS) which satisfies the condition 0 ≤ μ2A (x) + ν 2A (x) ≤ 1. If a decision maker (DM) gives the membership degree and the non-membership degree as 0.7 and 0.5, respectively, then this case is only valid for PFS. However, if a DM gives the membership degree and nonmembership degree as 0.7 and 0.8, respectively, then this case is valid for neither IFS nor PFS. As a result, the concept of PFS needs a further expansion. Yager [40] extended the concept of PFS to the concept of q-rung orthopair fuzzy set (q-ROFS) q q with the condition μ A (x) + ν A (x) ≤ 1 for each x ∈ X where q ≥ 1. Due to this useful structure, q-ROFSs have been often used to model uncertainties in decision making problems (see e.g., [8, 16, 36]). Most researchers are interested in the topological structure of fuzzy sets as well as real life applications. General topology deals with the special definitions given for spatial structure concepts, and it compares different definitions and investigates the connections between the structure-related properties defined on a set. Moreover, topology is concerned with the alternatives of a geometric object that are kept under continuous deformations, such as stretching, twisting, crumpling, and bending, but not tearing or gluing. A topological space is a set endowed with a topology, which allows defining continuous deformation of sub-spaces and a topology can be defined by using sets, continuous functions, manifolds, algebra, differentiable functions, differential geometry, etc. [11]. However, an ordinary topology cannot model uncertainties. In such a case, a fuzzy topology can be considered. Chang [3] defined and examined basic concepts and general properties of the fuzzy topological spaces. Also, Lowen [17, 18] studied the concepts of fuzzy compactness, initial and final fuzzy topologies. Then, structures of fuzzy topology has been extended for other fuzzy sets. For example, Çoker [5] proposed the concept of intuitionistic fuzzy topology and Olgun et al. [25] introduced the concept of Pythagorean fuzzy topology. Öztürk and Yolcu [27] presented some structures of PFSs such as interior, closure, base and sub-base. Türkarslan et al. [35] expanded the concept of Pythagorean fuzzy topology to the concept of q-rung orthopair fuzzy topology. Moreover, many researchers have studied the topological structures of various fuzzy sets (see, [7, 13, 19, 20, 30– 32]). However, due to lack of a proper fuzzification of point, one is unable to study convergence or other local properties. In order to avoid such inconveniences, Wong [38] defined the concept of fuzzy point. Moreover, many researchers studied on this concept (see e.g., [9, 14, 28]). With the introduction of the concept of fuzzy point, studies were started on the concept of neighborhood and convergence. Pao-Ming and Ying-Ming [28] introduced the concepts of quasi-coincidence and Q-neighbourhood for fuzzy points and defined Moore–Smith convergence in fuzzy topological spaces. Then the concept of fuzzy point and its properties were extended to several type of fuzzy points. For example, Çoker and Demirci [6] introduced the concept of intuitionistic fuzzy points, Lupianez [21] also studied on quasi-coincidence relations for intuitionistic fuzzy point. Lupianez [22] presented nets and filters in intuitionistic

14 q-Rung Orthopair Fuzzy Points and Applications …

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fuzzy topology. Olgun et al. [26] proposed the concept of Pythagorean fuzzy point and applied it to the pattern recognition. The purpose of the pattern recognition is to decide whether an object belongs to a pattern. The concepts of similarity measure and distance measure (information measures) for fuzzy sets are often used in the pattern recognition. A similarity measure is an important tool for measuring the degree of similarity between two fuzzy sets, while a distance measure is its opposite. Many researchers proposed similarity and distance measures between q-ROFSs using various methods to increase the sensitivity of the solution of the applications and used these measures to solve decision making problems. For example, Peng and Liu [29] proposed some information measures containing similarity and distance measures and applied them to pattern recognition, medical diagnosis and clustering. Wang et al. [37] introduced similarity measures based on cosine function while Liu et al. [15] introduced a cosine distance measure and carried out a decision process. However, most of the proposed similarity measures and distance measures are based on an average model or a weighted average model that omits the interaction among elements of a given universe. Therefore, they are not reasonable in some cases. It means an average or a weighted average model does not work well in many real life problems. In this chapter, to overcome this problem, we use the notion of Choquet integral that is a non-linear continuous aggregation function. Fuzzy measures are useful tools for handling interaction criteria in many situations [33]. The Choquet integral [4] that is an extended aggregation operator uses a fuzzy measure rather than an additive one. Compared to additive integrals such as the Lebesgue integral, the Choquet integral has a complicated structure. However, the Choquet integral is more successful in the aggregation process [23, 24]. Briefly, fuzzy measures and the Choquet integral allow us to consider the preferences that are not considered in additive aggregation operators such as the weighted arithmetic mean [34]. In this study, we define the notion of q-rung orthopair fuzzy point (q-ROFP) that is a generalization of the notion of Pythagorean fuzzy point [26] and propose a Dice similarity measure and a distance measure between q-ROFSs with the help of the Choquet integral. We also give an application on the pattern recognition by using q-ROFPs and the proposed Dice similarity. Moreover, we define the continuity of a function at a q-ROFP and the convergence of nets of q-ROFPs in q-rung orthopair fuzzy topological spaces. The remaining part of this chapter is organized as follows: In Sect. 14.2, we recall the concepts of q-ROFS and q-rung orthopair fuzzy topology. Then, we introduce the concept of q-ROFP and give some properties of q-ROFPs. In Sect. 14.3, we give a Dice similarity measure and a distance measure based on the Choquet integral between q-ROFSs. Then, we give an application on pattern recognition. In Sect. 14.4, we introduce the convergence of nets of q-ROFPs. We investigate the relationship between the convergence of nets and the continuity of functions defined between q-rung orthopair fuzzy topological spaces. In Sect. 14.5, we conclude the paper.

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14.2 q-Rung Orthopair Fuzzy Points Let q ≥ 1. A q-ROFS A of a universal set X is a pair (μ A , ν A ) of a membership function μ A : X → [0, 1] and a non-membership function ν A : X → [0, 1] with q q q μ A (x) + ν A (x) = r A (x) for any x ∈ X where the function r A : X → [0, 1] is called the strength of commitment [40]. Now, let us recall the set operations and topology over q-ROFSs : Definition 1 ([40]) Let A = (μ A , ν A ), B = (μ B , ν B ) be two q-ROFSs of a set X . Then, (i) the complement of A is defined by Ac := (ν A , μ A ), (ii) the intersection of A and B is defined by A ∩ B := (min {μ A , μ B } , max {ν A , ν B }), (iii) the union of A and B is defined by A ∪ B := (max {μ A , μ B } , min {ν A , ν B }), (iv) we say A is a subset of B or B contains A and we write A ⊂ B or B ⊃ A if μ A ≤ μ B and ν A ≥ ν B . Note here that, if the union and the intersection are infinite, then we use supremum and infimum, shortly ‘sup’ and ‘inf’, respectively. The concept of q-rung orthopair fuzzy topological space is proposed by Türkarslan et al. [35] as a generalization of Pythagorean fuzzy topological spaces [26]. Definition 2 ([35]) Let X be a non-empty set and let τ be a family of q-ROFSs of X . If (T1 ) 1 X , 0 X ∈ τ (T2 ) A 1 ∩ A2 ∈ τ , for any A1 , A2 ∈ τ Ai ∈ τ , for any index set I , (T3 ) i∈I

then τ is called a q-rung orthopair fuzzy topology on X and the pair (X, τ ) is called a q-rung orthopair fuzzy topological space where 1 X and 0 X are q-ROFSs that are given with (1, 0) and (0, 1), respectively. It is clear that a 2−rung orthopair fuzzy topology is actually a Pythagorean fuzzy topology. Now, we are ready to introduce the notion of q-ROFP and some related concepts. Before we recall the image and pre-image of a q-ROFS with respect to a function. Throughout this paper, we use the notation “ g[A] ” for the image and “ g −1 [B] ” for pre-image of a q-ROFSs A and B, respectively, where g is a function. Definition 3 ([35]) Let X and Y be two non-empty sets, let g : X → Y be a function and let A and B be q-ROFSs of X and Y , respectively. Then, the membership and the non-membership functions of image of A are defined by ⎧ μ A (z) , if g −1 (y) = ∅ ⎪ ⎨ z∈gsup −1 (y) μg[A] (y) := ⎪ ⎩ 0, otherwise

14 q-Rung Orthopair Fuzzy Points and Applications …

and νg[A] (y) :=

249

⎧ μ A (z) , if g −1 (y) = ∅ ⎪ ⎨ z∈ginf −1 (y) ⎪ ⎩

1,

otherwise,

respectively. The membership and the non-membership functions of pre-image of B are defined by μg−1 [B] (x) := μ B (g(x)) and νg−1 [B] (x) := ν B (g(x)), respectively. Note that g[A] and g −1 [B] are q-ROFSs (see, [35]). The following proposition is a generalization of Proposition 1 of [26] for q-ROFSs. Proposition 1 Let X and Y be two non-empty sets, let g : X → Y be a function, and let {Ai : i ∈ I } be an arbitrary collection of q-ROFSs of X . Then, we have  g



=

Ai

i∈I



g(Ai ).

(14.1)

i∈I

Proof Let y ∈ X . If g −1 (y) = ∅ then μ

g



Ai

 (y)

= 0 and ν

i∈I

g



i∈I

Ai

 (y)

= 1. On the

other hand, since μg[Ai ] (y) = 0 and νg[Ai ] (y) = 1, for any i ∈ I we have μ  g[Ai ] (y) = sup μg[Ai ] (y) = 0 i∈I

i∈I

and

ν  g[Ai ] (y) = inf νg[Ai ] (y) = 1. i∈I

i∈I

If g −1 (y) = ∅, then we obtain μ

g



i∈I

Ai

 (y)

= sup μ  Ai (z) z∈g −1 (y)

i∈I

= sup sup μ Ai (z) z∈g −1 (y) i∈I

= sup sup μ Ai (z) i∈I z∈g −1 (y)

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= sup μg[Ai ] (y) i∈I 

=μ

g[Ai ] (y)

i∈I

and

ν

g



Ai

 (y)

i∈I

= inf

z∈g −1 (y)

= inf

ν  Ai (z) i∈I

inf ν Ai (z)

z∈g −1 (y) i∈I

= inf

inf

i∈I z∈g −1 (y)

ν Ai (z)

= inf νg[Ai ] (y) i∈I

=ν  g[Ai ] (y). i∈I



Therefore, the proof is completed. Now, we define the concept of q-ROFP.

Definition 4 Let X be a set, let u ∈ X , let r ∈ [0, 1], and α ∈ [0, 1) and β ∈ (0, 1] such that α q + β q = r q with q ≥ 1. Then the q-ROFS [u]α,β = (cα , 1 − c1−β ) is called a q-ROFP in X where α, x = u cα (x) := 0, x = u. It is obvious that (1 − c1−β )(x) :=

β, x = u 1, x = u.

Here, the function r[u]α,β : X → [0, 1] given with r, x = u r[u]α,β (x) := 1, x = u is called the strength of commitment of the q-ROFP [u]α,β . We simply use [u] when there is no confusion. The ordinary point u is called the support of [u]. Definition 5 Two q-ROFPs [u 1 ]α1 ,β1 and [u 2 ]α2 ,β2 are said to be equal if u 1 = u 2 , α1 = α2 and β1 = β2 . In this case, we write [u 1 ] = [u 2 ]. A q-ROFP [u]α,β in X is said to be contained in a q-ROFS A of X if α ≤ μ A (u) and β ≥ ν A (u).

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To avoid confusion we use the notation u ∈ X whenever u is an ordinary point in X and we write [u] ∈ X whenever [u] is a q-ROFP in X . Following two propositions can be proved by conducting the similar idea in Propositions 2 and 3 of [26]. Proposition 2 A q-ROFS A is expressed as the union of all the q-ROFPs which are contained by A. Proposition 3 Let g : X → Y be a function and let [u]α,β ∈ X . Then the image of [u]α,β is again a q-ROFP in Y with the support g(u) and the strength of commitment r g[[u]α,β ] = r[u]α,β and g[u] := g[[u]α,β ] = [g(u)]α,β .

14.3 A Pattern Recognition Application In this section, first we introduce a Dice similarity measure and a distance measure based on Choquet integral between q-ROFPs and then we give an application on the pattern recognition.

14.3.1 A Dice Similarity Measure and A Distance Measure First, we recall an existing Dice similarity and a distance measure between q-ROFSs. Let X = {x1 , ..., xn } be a finite set and let A = (μ A , ν A ) and B = (μ B , ν B ) be two q-ROFSs of X . A Dice similarity measure [12] and a distance measure [15] between A and B are given by

D Wq−R OFS

(A, B) :=

n

i=1

wi 

 q  q q q 2 μ A (xi )μ B (xi ) + ν A (xi )ν B (xi ) 2q

2q

2q

2q

μ A (xi ) + ν A (xi ) + μ B (xi ) + ν B (xi )



(14.2)

and 

1 n 2  q 2 2 1  q q q (A, B) := ωi μ A (xi ) − μ B (xi ) +  ν A (xi ) − ν B (xi )  2 i=1 (14.3) where w = (w1 , ..., wn ) is the weight vector with w j ∈ [0, 1] for all j = 1, ..., n such n

w j = 1. that Dis Wq−R OFS

j=1

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To introduce a new Dice similarity measure and distance measure we use the Choquet integral. The basis of this integral is inherently fuzzy measure. Therefore, first we recall the concept of fuzzy measure. Definition 6 ([4, 33]) Let X be a finite set and let P(X ) be the power set of X . If, (i) σ (∅) = 0, (ii) σ (X ) = 1, (iii) A ⊆ B implies σ (A) ≤ σ (B) (monotonicity), then the set function σ : P(X ) → [0, 1] is called a fuzzy measure on X . Definition 7 ([4]) Let X = {x1 , x2 , ..., xn } be a finite set and let σ be a fuzzy measure on X . The Choquet integral of a function f : X → [0, 1] with respect to σ is defined by  (C)

f dσ := X

n



 f (x(k) ) − f (x(k−1) ) σ (E (k) ),

(14.4)

k=1

 n where the sequence x(k) k=0 is the permutation of the sequence {xk }nk=0 such that   0 := f (x(0) ) ≤ f (x(1) ) ≤ f (x(2) ) ≤ ... ≤ f (x(n) ) and E (k) := x(k) , x(k+1) , ..., x(n) . Now, we are ready to propose a Dice similarity measure and distance measure based on Choquet integral between q-ROFSs by modifying (14.2) and (14.3). Definition 8 Let X = {x1 , ..., xn } be a finite set and let A = (μ A , ν A ) and B = (μ B , ν B ) be two q-ROFSs in X and let σ be a fuzzy measure on X . A Choquet Dice similarity measure between A and B (with respect to the fuzzy measure σ ) is given with  D−(C) (1) Wq−R O F S (A, B) := (C) f A,B dσ (14.5) X

where

(1) f A,B (xi )

:=

 q  q q q 2 μ A (x i )μ B (x i ) + ν A (x i )ν B (x i ) , 2q 2q 2q 2q μ A (xi ) + ν A (xi ) + μ B (xi ) + ν B (xi )

for i = 1, ..., n.

D−(C) Proposition 4 The Choquet Dice similarity measure Wq−R O F S satisfies the following properties: D−(C) (P1 ) 0 ≤ Wq−R O F S (A, B) ≤ 1, D−(C) D−(C) (P2 ) Wq−R O F S (A, B) = Wq−R O F S (B, A), D−(C) (P3 ) A = B if and only if Wq−R O F S (A, B) = 1, for any q-ROFSs A and B.  q 2  q 2 q q (1) (xi ) Proof (P1 ) Since μ A (xi ) − μ B (xi ) + ν A (xi ) − ν B (xi ) ≥ 0, we have f A,B ∈ [0, 1] for i = 1, ..., n. Therefore, from the monotonicity of the Choquet integral

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D−(C) we obtain 0 ≤ Wq−R O F S (A, B) ≤ 1. (1) (1) (xi ) = f B,A (xi ) for i = 1, ..., n. (P2 ) It is trivial since f A,B q q q q (P3 ) If A = B then μ A (xi ) = μ B (xi ) and ν A (xi ) = ν B (xi ) for i = 1, ..., n. Thus, we (1) D−(C) D−(C) have f A,B (xi ) = 1 and so Wq−R O F S (A, B) = 1. Conversely, let Wq−R O F S (A, B) =  2 q q (1) (xi ) = 1 for all i = 1, .., n. It means that μ A (xi ) − μ B (xi ) + 1. Therefore, f A,B  q 2 q ν A (xi ) − ν B (xi ) = 0 and so we obtain A = B. Thus, the proof is completed. 

Definition 9 Let X = {x1 , ..., xn } be a finite set and let A = (μ A , ν A ) and B = (μ B , ν B ) be two q-ROFSs in X and let σ be a fuzzy measure on X . A Choquet distance measure between A and B (with respect to the fuzzy measure σ ) is given with ⎛ Dis−(C) Wq−R OFS

(A, B) := ⎝(C)



⎞ 21 (2) f A,B

dσ ⎠

(14.6)

X

where q ≥ 1.

(2) f A,B (xi )

   q  μ (xi ) − μq (xi )2 +  ν q (xi ) − ν q (xi ) 2 A B A B , for i = 1, ..., n and := 2

Dis−(C) Proposition 5 The Choquet distance measure Wq−R O F S satisfies the following properties: Dis−(C) (P1 ) 0 ≤ Wq−R O F S (A, B) ≤ 1, Dis−(C) Dis−(C) (P2 ) Wq−R O F S (A, B) = Wq−R O F S (B, A), Dis−(C) (P3 ) A = B if and only if Wq−R O F S (A, B) = 0. Dis−(C) Dis−(C) (P4 ) If C is a q-ROFS in X and A ⊂ B ⊂ C then Wq−R O F S (A, B) ≤ Wq−R O F S Dis−(C) Dis−(C) (A, C) and Wq−R O F S (B, C) ≤ Wq−R O F S (A, C) for any q-ROFSs A, B and C.

Proof 0 ≤ μ A (xi ), μ B (xi ), i ) ≤ 1 for i = 1, ..., n, we have  q(P1 ) Since ν Aq (xi ), ν A (x 2 2 q q    0 ≤ μ A (xi ) − μ B (xi ) ≤ 1 and 0 ≤ ν A (xi ) − ν B (xi )  ≤ 1 for i = 1, ..., n and (2) so we get f A,B (xi ) ∈ [0, 1]. Since the Choquet integral is monotone, we obtain Dis−(C) 0 ≤ Wq−R O F S (A, B) ≤ 1. (2) (2) (xi ) = f B,A (xi ) for i = 1, ..., n. (P2 ) It is trivial since f A,B q q q q (P3 ) If A = B then μ A (xi ) = μ B (xi ) and ν A (xi ) = ν B (xi ) for i = 1, ..., n. Thus, we  q   q  q q (2) (xi ) = have μ A (xi ) − μ B (xi ) = 0 and  ν A (xi ) − ν B (xi )  = 0. As a result of f A,B Dis−(C) Dis−(C) 0 we get Wq−R O F S (A, B) = 0. Conversely, let Wq−R O F S (A, B) = 0. This implies q q q (2) (xi ) = 0 for i = 1, .., n. Thus, we have μ A (xi ) = μ B (xi ) and ν A (xi ) = that f A,B q ν B (xi ) for i = 1, ..., n. It yields that A = B.

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q

q

q

q

q

(P4 ) If A ⊂ B ⊂ C then μ A (xi ) ≤ μ B (xi ) ≤ μC (xi ) and ν A (xi ) ≥ ν B (xi ) ≥ νC (xi ), for i = 1, ..., n. Thus, we have   q μ (xi ) − μq (xi )2 A B   q μ (xi ) − μq (xi )2 B C   q ν (xi ) − ν q (xi )2 A B   q ν (xi ) − ν q (xi )2 B C

 q 2 q ≤ μ A (xi ) − μC (xi )  q 2 q ≤ μ A (xi ) − μC (xi )  q 2 ≤ ν A (xi ) − νC2 (xi )  q 2 q ≤ ν A (xi ) − νC (xi ) .

(2) (2) (2) (2) So, we obtain f A,B (xi ) ≤ f A,C (xi ) and f B,C (xi ) ≤ f A,C (xi ) which yield with the Dis−(C) Dis−(C) monotonicity of the Choquet integral that Wq−R O F S (A, B) ≤ Wq−R O F S (A, C) and Dis−(C) Dis−(C)  Wq−R O F S (B, C) ≤ Wq−R O F S (A, C). Hence, the proof is completed.

14.3.2 Pattern Recognition In this sub-section we give a pattern recognition application of the proposed Choquet Dice similarity measure via q-ROFPs. Example 1 Let A1 , A2 and A3 be three patterns which are represented by following 3−ROFPs with the same support x1 of the universal set X = {x1 , x2 , x3 } : A1 = [x1 ]0.7,0.8 = { x1 , 0.7, 0.8 , x2 , 0.0, 1.0 , x3 , 0.0, 1.0} A2 = [x1 ]0.5,0.9 = { x1 , 0.5, 0.9 , x2 , 0.0, 1.0 , x3 , 0.0, 1.0} A3 = [x1 ]0.9,0.6 = { x1 , 0.9, 0.6 , x2 , 0.0, 1.0 , x3 , 0.0, 1.0} . Let A = [x1 ]0.85,0.6 = { x1 , 0.85, 0.6 , x2 , 0.0, 1.0 , x3 , 0.0, 1.0} be a pattern that needs to be classified in one of three classes A1 , A2 and A3 . Now, considering the hypothetical fuzzy measure given in Table 14.1, we calculate the similarities with D−(C) respect to Wq−R OFS . [x1 ]0.7,0.8 ,

f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x1 ) ≤ f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x2 ) = f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x3 ) we get Since

Table 14.1 Fuzzy measure σ (∅) = 0 σ ({x3 }) = 0.3 σ ({x2 , x3 }) = 0.8

σ ({x1 }) = 0.4

σ ({x2 }) = 0.3

σ ({x1 , x2 }) = 0.6 σ ({x1 , x2 , x3 }) = 1

σ ({x1 , x3 }) = 0.7

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Table 14.2 Comparison of classification results of the Dice similarity measures in Example 1 Dice similarity Similarity Scores measures (A , A) (A , A) (A , A) 1

2

3

D Wq−R OFS

0.9197

0.7930

0.9947

D−(C)

0.9598

0.8965

0.9973

Wq−R O F S

 D−(C) Wq−R O F S ([x 1 ]0.7,0.8 , [x 1 ]0.85,0.6 ) = (C)

f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 dσ X

3 

 f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x(k) ) − f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x(k−1) ) σ (E (k) ) k=1   = f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x1 ) − f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x0 ) σ (E (1) )   + f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x2 ) − f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x1 ) σ (E (2) )   + f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x3 ) − f [x1 ]0.7,0.8 , [x1 ]0.85,0.6 (x2 ) σ (E (3) ) =

= (0.7994 − 0) × 1 + (1 − 0.7994) × 0.8 + (1 − 1) × 0.3 = 0.9598. D−(C) With similar calculations we can also calculate Wq−R O F S (A, Ai ) for i = 2, 3 (see, Table 14.2). According to the recognition principle of maximum degree of similarity between q-ROFSs, the process of assigning the unknown pattern A to Ai is described by   D−(C) (14.7) k = arg max Wq−R O F S (A, Ai ) . 1≤i≤4

The results in Table 14.2 show that pattern A belongs to A3 . Note that, ωi = σ ({xi }) D in the calculation of Wq−R OFS.

14.4 Continuity and Convergence In this section, we deal with the convergence of the nets of q-ROFPs and continuity of functions defined between q-rung orthopair fuzzy topological spaces. First, we recall some basic definitions. Definition 10 ([35]) Let A, U be two q-ROFSs in a q-rung orthopair fuzzy topological space. If there exists an open q-ROFS F such that A ⊂ F ⊂ U then U is called a neighbourhood of A.

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Proposition 6 [35] A q-ROFS A is open in a q-rung orthopair fuzzy topological space if and only if it contains a neighbourhood of its each subset. Definition 11 ([35]) Let (X, τ1 ) and (Y, τ2 ) be two q-rung orthopair fuzzy topological spaces and let g : X → Y be a function. If for any q-ROFS A of X and for any neighbourhood V of g[A] there exists a neighbourhood U of A such that g[U ] ⊂ V then g is called (q-rung orthopair fuzzy) continuous. Now, we introduce the concept of q-rung orthopair fuzzy continuity of a function at a q-ROFP. Definition 12 Let (X, τ1 ) and (Y, τ2 ) be two q-rung orthopair fuzzy topological spaces and let g : X → Y be a function. Then, g is said to be q-rung orthopair fuzzy continuous at a q-ROFP [u] if for any q-rung orthopair fuzzy neighbourhood V of g[u] there exists a q-rung orthopair fuzzy neighbourhood U of [u] such that f [u] ⊂ V. The following theorem gives the relationship between q-rung orthopair fuzzy continuity everywhere and q-rung orthopair fuzzy continuity at a q-ROFP that can be proved as Theorem 2 of [26]. Theorem 1 Let (X, τ1 ) and (Y, τ2 ) be two q-rung orthopair fuzzy topological spaces and let g : X → Y be a function. Then, g is q-rung orthopair fuzzy continuous everywhere if and only if it is q-rung orthopair fuzzy continuous at each q-ROFP of X. Now, we introduce the concept of q-rung orthopair fuzzy net. Definition 13 Let S be a directed set (see, e.g. [2]). A function from S to the set of q-ROFSs of X is called a q-rung orthopair fuzzy net in X and it is given with   {[u t ]}t∈S =: [u t ]αt ,βt t∈S .

(14.8)

We need the following definitions to define the concept of convergence of a q-rung orthopair fuzzy net to a q-ROFP. Definition 14 A q-ROFP [u]α,β is said to be quasi-coincident with a q-ROFS A, α + μ A (u) > 1 and β + ν A (u) < 1. In this case, we write [u]α,β q A or briefly [u]q A. Example 2 X = {u 1 , u 2 } be a finite set and let A = { u 1 , 0.5, 0.1 , u 2 , 0.6, 0.9} and B = { u 1 , 0.1, 0.2 , u 2 , 0.4, 0.0} be two 3−ROFSs of X. Let us consider the 3−ROFP [u 1 ]0.6,0.1 . It is obvious that [u 1 ]0.6,0.1 q A and [u 1 ]0.6,0.1 is not quasi-coincident with B.

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Definition 15 A fuzzy set A in (X, τ ) is called a Q-neighbourhood of a q-ROFP [u] if there exists an open q-ROFS B ⊂ A such that [u]q B. The family of all Qneighbourhoods of [u] is called the Q-neighbourhood system of [u] and denoted by N q[u]. Olgun et al. [26] defined the convergence of nets in Pythagorean fuzzy topological spaces. Now, we introduce the convergence of nets in q-rung orthopair fuzzy topological spaces by motivating from [26] and [28]. Definition 16 A q-rung orthopair fuzzy net {[u t ]}t∈S in (X, τ ) is said to be convergent to a q-ROFP [u] if for any Q-neighbourhood A of [u] there exists t0 ∈ S such that [u t ]q A whenever t ≥ t0 . In this case, we write lim[u t ] = [u]. t∈S

Now, we introduce another version of q-rung orthopair fuzzy continuity at a q-ROFP by using Q-neighbourhoods. Definition 17 Let (X, τ1 ) and (Y, τ2 ) be two q-rung orthopair fuzzy topological spaces and let g : X → Y be a function. Then, g is said to be Q-q-rung orthopair fuzzy continuous at a q-ROFP [u] if for any Q-neighbourhood V of g[u] there exists a Q-neighbourhood U of [u] such that g[U ] ⊂ V . The following theorem shows that the concept of Q-q-rung orthopair fuzzy continuity at a q-ROFP is more general than the concept of q-rung orthopair fuzzy continuity and it can be proved as Theorem 3 of [26]. Theorem 2 Let (X, τ1 ) and (Y, τ2 ) be two q-rung orthopair fuzzy topological spaces and let g : X → Y be a function. If g is q-rung orthopair fuzzy continuous everywhere, then it is Q-q-rung orthopair fuzzy continuous at each q-ROFP [u] of X . The following theorem gives the relationship between Q-q-rung orthopair fuzzy continuity at a q-ROFP and convergence of nets and it can be proved as Theorem 4 of [26]. Theorem 3 Let (X, τ1 ) and (Y, τ2 ) be two q-rung orthopair fuzzy topological spaces. If a function g : X → Y is Q-q-rung orthopair fuzzy continuous at a point [u] then lim g[u t ] = g[u] for any net [u t ]αt ,βt such that lim[u t ]αt ,βt = [u]α,β . t∈S

t∈S

All of the definitions and results given in Sect. 14.4 are generalizations of definitions and results given in subsection 4.2 of [26] for q ≥ 2 instead of q = 2 in the Pythagorean case.

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14.5 Conclusion In this chapter, we introduce the concept q-rung orthopair fuzzy point. Then, we propose a Dice similarity measure and a distance measure based on the Choquet integral and give an application to pattern recognition. We also define the continuity of a function at a q-rung orthopair fuzzy point that is defined between two q-rung orthopair fuzzy topological spaces. Moreover, we present the convergence of nets in q-rung orthopair fuzzy topological spaces. In the future studies, compactness in q-rung orthopair fuzzy topologies can be studied and results obtained in this study can be studied for interval valued fuzzy topologies.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

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