Proximity Approach to Problems in Topology and Analysis 9783486598605, 9783486589177

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Proximity Approach to Problems in Topology and Analysis von Somashekhar Naimpally

Oldenbourg Verlag München

Somashekhar Naimpally is an Emeritus Professor at Lakehead University, Thunder Bay, Canada. He studied at the University of Bombay, Mumbai, India and Michigan State University, East Lansing, USA. He has coauthored books on Proximity Spaces, Symmetric Generalized Topological Structures and Leelavati.

© 2009 Oldenbourg Wissenschaftsverlag GmbH Rosenheimer Straße 145, D-81671 München Telefon: (089) 45051-0 oldenbourg.de All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the Publishers. Editor: Kathrin Mönch Producer: Dr. Rolf Jäger Cover design: Kochan & Partner, München Printed on acid-free and chlorine-free paper Printing: Books on Demand GmbH, Norderstedt ISBN 978-3-486-58917-7

Preface The book Proximity Spaces by S. A. Naimpally and B. D. Warrack (Cambridge Tracts in Mathematics and Mathematical Physics No. 59, Cambridge University Press, 1970) dealt primarily with Efremovič proximity. The generalized proximity was discovered jointly by Leader (non-symmetric) and his student Lodato (symmetric). The symmetric case was studied comprehensively by Mozzochi and this material is contained in the book C. J. Mozzochi, M. S. Gagrat and S. A. Naimpally, Symmetric Generalized Topological Structures (Exposition Press, Hicksville, New York, 1976). Various other proximities were discovered and applied to many problems in topology and analysis. With the suggestion and encouragement from my wife Sudha, I decided to write this book. Most of the material in this book contains work done in collaboration with several colleagues and I am most grateful to them. In the above-mentioned book, the material principally concerned Efremovič proximity. After its publication, mathematicians discovered various other proximities and applied them to many problems in topology. In this book, we propose to introduce various proximities and their use in solving problems in topology and analysis. Since this book is meant for research mathematicians, we omit proofs of statements that are straightforward. We are primarily interested in applications; we have refrained from giving results, constructions or counter examples that are complicated. Some of them are in the earlier book and for others we give references where an interested reader can find the relevant material. Chapters 7, 8, 9, 11, 12 and 13 contain material related to problems in analysis of Brown, Bartle, Pettis, Kelley, Weston and others. As with the earlier book Proximity Spaces, the present book presupposes that the reader is familiar with basic topology as is contained in standard texts such as General Topology by John L. Kelley; this text is our principal reference. I take this opportunity to express my gratitude to my teachers in India: Professors N. H. Phadke, D. P. Patravali, G. V. Bhagawat, P.R. Masani; in USA: Professor J. G. Hocking who first showed me the excitement of research and Professor D. E. Sanderson who nurtured it. Thanks are due to my recent collaborators Anna Di Concilio, Giuseppe (Peppe) Di Maio, late Enrico Meccariello, Jeff Mozzochi. I am indebted to Anna and Peppe for their kind hospitality and for getting research grants for over 30 visits to Italy from the National Research Council of Italy. Peppe also organized an International Conference on my 70th birthday in 2001.

VI

Preface

Rajan Anantharaman, Jeff Mozzochi, Peppe Di Maio, Ellen Reed, Horst Herrlich, Harry Poppe and René Bartsch made many valuable suggestions. I am grateful for René’s invaluable help in many ways. Anantharaman, Peppe, Mohan Lal Tikoo also helped with proof reading and removal of obscurities from the earlier drafts. Jesús Rodríguez López provided references with notes for quasi-proximities. I am grateful for their help.

Somashekhar (Som) Naimpally [email protected]

Foreword “If a result has value it is when, by binding together long known elements, until now scattered and appearing unrelated to each other, it suddenly brings order where there reigned apparent disorder.” - Henri Poincaré ICM, Roma (1908)

This book shows the simplicity and the power of various proximities. It shows how proximities and their generalization nearness are useful in dealing with T1 and Hausdorff compactifications, extensions of continuous functions from dense subspaces, hyperspace topologies, function spaces, unified approach to metrisation theorems, semi-metric and developable spaces with application to Relativity, Open and uniformly open relations occurring in functional analysis, generalization of duality in functional analysis etc. Proximities lie hidden in many parts of analysis/topology. When one looks at concepts and results from this new angle, they appear simpler, and provide shorter and more transparent proofs in place of the long calculative ones. The tract Proximity Spaces deals primarily with proximity discovered by Efremovič which was simply called “proximity”. The present book begins with properties of Leader-Lodato proximity which is referred to as Lodato or LO-proximity in the above Tract and as “symmetric generalized proximity” in the book by Mozzochi-Gagrat-Naimpally. Occasionally we use, especially in informal discussions, the word proximity to mean a nearness relation between pairs of sets. After studying the basic properties and introducing the tools of bunches and clusters, a construction of a T1 compactification of a T1 topological space is given. The construction parallels that of the well-known Wallman compactification and includes all compactifications such as Alexandroff, Wallman, Wallman-Frink, Freudenthal, Stone-Čech, and Smirnov as special cases. The first chapter ends with Mozzochi uniformity, which has played an important role in the solution of a problem posed by the AMS Veblen Prize winner Brown concerning developable spaces; Domiaty and Laback have found it useful in the study of a paper on General Relativity by Hawking-King-McCarthy. The second chapter studies the Efremovič proximity and the Smirnov compactification which generalizes all Hausdorff compactifications. Then follows the relationship between the proximity and compatible uniformities. It concludes with an application to Hahn-Banach Theorem in a non-compact topological-convexity space.

VIII

Foreword

There is a vast literature on continuous extensions of continuous functions from dense subspaces. In the third chapter a generalization of Taimanov theorem is proved using the tools of LO-proximity spaces. The domain has a LO-proximity and the range has Efremovič proximity.This result includes all special results in this area; its applications include comparisons of Hausdorff compactifications of a Tychonoff space. Nearness, a generalization of proximity and (Weil) uniformity was discovered by Herrlich and it is studied in Chapter 4. Using nearness, a further generalization of the Taimanov theorem is constructed which includes many non-compact extensions due to Banashewski, Katetov and others. A simple result on continuous extensions of continuous functions from dense subspaces generalizes results scattered in many research articles. Using this result, it is child’s play to prove results involving reflective functors. The main idea is simple: replace intersection by near. In topology ultrafilters play a vital role but they yield only one extension or a compactification. By using clusters or bunches which are analogues of ultrafilters with near replacing intersection, one gets all extensions without much further effort! The categorical point-of-view concludes this chapter; only basic information is given since there are articles in which details are available.. Chapter 5 deals with hyperspace topologies. Basic information is given about the well known hypertopologies: Vietoris, Hausdorff-metric, LO-proximal, Wijsman, Fell, locally finite, uniformly discrete, Poppe Delta, U and Hausdorff-Bourbaki. The treatment is simplified by LO-proximity and includes the place of the above hypertopologies from the lattice theoretic point of view. We demonstrate the importance of proximities in the study of hypertopologies in Chapter 6. A simple characterization of the Wijsman hypertopology in terms of two proximities leads not only to a generalization to T1 spaces, but also helps in proving results in a few lines; these results need long calculative proofs in the setting of metric spaces. Using two compatible LO-proximities, a hypertopology, called Bombay topology, is studied which includes all known hypertopologies as special cases. Results involving it contain literature scattered in numerous papers. For example, one result, which compares two Bombay topologies, yields over a hundred special cases involving comparisons of two hyperspace topologies. This is due to clarity, hindsight, emerging over the years. In measure theory on a metric space, one needs a hypertopology which is independent of the metric. This leads to the problem of sup and inf of Hausdorff metric topologies on the hyperspace of a metric space. In Chapter 7, this problem is studied in the general case of Hausdorff-Bourbaki uniformities. The use of proximity yields proofs which are much simpler than those that appeared initially in the metric case. In classical complex analysis, continuous convergence is used in dealing with function spaces of analytic functions. In Chapter 8, we study a generalization to hypertopologies. Only the cases of Hausdorff-metric, Vietoris, LO-proximal and Fell hypertopologies are discussed. The results are interesting and throw light on the earlier results. In the study of function space topologies, the compact-open topology plays a vital role. In Chapter 9, we study its generalization to LO-proximal-set-open topology and its relation-

Foreword

IX

ships with the well-known function space topologies such as the pointwise convergence, the group topology associated with the homeomorphism groups of a topological space. There are three types of metrisation results in topology, viz. topological, proximal and uniform. There is a belief that they are of a different genre from each other. Contrary to this view, we give a unified treatment in Chapter 10 with the use of proximity. A panoramic view of important metrisation theorems is given. Developable spaces are important in topology and play a vital role in the metrisation theorem due to Bing; they lie midway between general topological spaces and metrisable spaces. With the help of LO-proximity and Mozzochi uniformity, a solution to a problem posed by Brown is given in Chapter 11. This result has proven to be useful in dealing with a paper in general relativity by Hawking-King-McCarthy. There is also some material on various types of maps between topological spaces. In functional analysis, open and uniformly open maps play a significant role. Many mathematicians, including Kelley, Pettis, and Weston, have studied such maps in metric and uniform spaces. A simple result involving proximity is used in Chapter 12 to solve a problem posed by Pettis and also generalize some of the known results with simplified proofs. Duality plays an important role in functional analysis but there is very little literature in General Topology. Using a technique of Bartle, Chapter 13 contains duals of equicontinuity, even continuity and some other related concepts recently discovered by Bouziad and Trolliac. Simple proofs of well known results, including the Ascoli-Arzelá Theorem are given. Results in topological groups of homeomorphisms due to Arens, Dieudonné, Di Concilio are generalized and simpler proofs are given using LO-proximal-set-open topologies. In Chapter 14 we study uniform invariants in terms of proximities. In general, Efremovič proximity has a unique compatible totally bounded uniformity as well as uniformities that are not totally bounded. So one cannot characterize total boundedness with one proximity. We show that one can characterize each of total boundedness, completeness, compactness by a pair of proximities. The setting is in metric spaces. I have tried to make the book reader friendly in the following way. Definitions are given just before their use and not bundled together. Terms, well known to topologists, but not known to others are defined. Many important definitions are repeated so that the reader is not forced to look for them. Along with every symbol its meaning in words is also given. Each chapter begins with an abstract with Mathematics Subject Classification 2000 (MSC) numbers to aid the readers to locate further references. We use the standard abbreviations: iff for “if and only if” and for the end of a proof or a discussion. In the tract Proximity Spaces, the word “proximity” referred to the one discovered by Efremovič; in 1960’s that was the only proximity that was studied extensively. But now there are many proximities which have found use in applications. So in this book “proximity” refers to any of the proximities mentioned in this book. The above proximity is called “Efremovič proximity”; others are given appropriate adjectives such as Lodato or LO-proximity, regular proximity etc.

Contents Preface

V

Foreword

VII

Intuitive Introduction

1

1

The Lodato Proximity

3

1.1

Introduction ................................................................................................................ 4

1.2

Tools: Clusters and Bunches .................................................................................... 13

1.3

A General T1 Compactification ................................................................................ 16

1.4

Mozzochi Uniformity and LO-Proximity................................................................. 21

2

The Efremovič Proximity

2.1

Introduction .............................................................................................................. 25

2.2

Tools: Clusters.......................................................................................................... 29

2.3

Smirnov Compactification ....................................................................................... 31

2.4

Tychonoff Topology, Efremovič Proximity and Uniformity ................................... 33

2.5

Appendix: Application of Wallman-Frink Compactification................................... 36

3

Extensions of Continuous Functions from Dense Subspaces

3.1

Generalized Taimanov Theorem .............................................................................. 39

3.2

Hausdorff Compactifications ................................................................................... 42

3.3

Applications of the Generalized Taimanov Theorem............................................... 43

3.4

More Applications.................................................................................................... 46

4

Nearness and Contiguity. Extension Theorems

4.1

Introduction .............................................................................................................. 50

4.2

Extension Theorems ................................................................................................. 53

4.3

Topological Properties in Terms of Nearness .......................................................... 56

25

39

49

XII

Contents

4.4

Categorical Point of View.........................................................................................58

5

Hyperspace Topologies

5.1

What Is a Hit-and-Miss Topology? An Intuitive Introduction .................................61

5.2

Vietoris Proximal and Fell Topologies .....................................................................64

5.3

Hausdorff Metric Topology ......................................................................................67

5.4

Wijsman Convergence Topology..............................................................................70

5.5

Locally Finite Topology............................................................................................73

5.6

Ball and Proximal Ball Topologies ...........................................................................75

5.7

Poppe’s Delta Topologies .........................................................................................78

5.8

Hausdorff-Bourbaki Uniformity ...............................................................................79

5.9

Uniformly Discrete Hypertopology ..........................................................................83

5.10

U Topology ...............................................................................................................85

5.11

Bounded Topologies .................................................................................................85

6

Hit-And-Far-Miss Topology: A Generalization of the Wijsman Topology

6.1

Introduction...............................................................................................................87

6.2

Comparisons among the three topologies .................................................................88

6.3

Comparisons with Other Topologies.........................................................................91

6.4

Appendix I: Relative Total Boundedness..................................................................92

6.5

Appendix II: Bombay Topology, a Unification ........................................................93

7

Infimum and Supremum of Hypertopologies

7.1

Inf of Hausdorff-Bourbaki Uniformities ...................................................................97

7.2

Sup of Hausdorff-Bourbaki Uniformities..................................................................99

7.3

Sup of Wijsman Topologies....................................................................................100

8

Hyper-Continuous Convergence in Function Spaces

8.1

Introduction.............................................................................................................103

8.2

Hausdorff-Continuous Convergence.......................................................................105

8.3

Vietoris convergence...............................................................................................108

8.4

Proximal-Continuous-Convergence ........................................................................109

8.5

Fell-Continuous-Convergence ................................................................................110

61

87

97

103

Contents

XIII

9

Proximal Set-Open Topology

113

9.1

Introduction ............................................................................................................ 113

9.2

Comparison of Proximal Set-Open Topology with Other Topologies ................... 118

9.3

Separation Axioms ................................................................................................. 118

10

A Unified Approach to Metrisation Problems

121

11

Semi-Metric and Developable Spaces

125

11.1

Introduction ............................................................................................................ 125

11.2

Developable Spaces................................................................................................ 127

11.3

Metrisable Spaces................................................................................................... 129

11.4

Mappings................................................................................................................ 130

12

Open and Uniformly Open Relations

133

13

Duality in Function Spaces

139

13.1

Introduction ............................................................................................................ 139

13.2

Equicontinuity ........................................................................................................ 143

13.3

Even Continuity...................................................................................................... 144

13.4

Topological Groups of Homeomorphisms ............................................................. 147

13.5

Generalizations of Equicontinuity.......................................................................... 151

13.6

Application ............................................................................................................. 153

14

Proximal and Other Characterizations of Uniform Invariants

14.1

Completeness ......................................................................................................... 155

14.2

Total Boundedness ................................................................................................. 157

14.3

UC Spaces .............................................................................................................. 158

14.4

CU Spaces .............................................................................................................. 160

14.5

Compactness........................................................................................................... 160

15

Applications

155

163

15.1 General Relativity .................................................................................................. 163 References ............................................................................................................................ 166 15.2 Differential Equations and Mathematical Economics ............................................ 167 References ............................................................................................................................ 170 15.3 Digital Images and Sound Analysis & Synthesis ................................................... 171 References ............................................................................................................................ 176

XIV Bibliography

Contents 177

References for quasi-proximities...........................................................................................192 Basic facts .............................................................................................................................192 Cardinality of compatible quasi-proximities and of a quasi-proximity class ........................194 Completions and compactifications ......................................................................................196 Hypertopologies and functions spaces ..................................................................................198 Fuzzy topology......................................................................................................................199 Index

201

Intuitive Introduction* “The significant problems we face cannot be solved by the same level of thinking that created them.” - Albert Einstein

In this section, we propose to give a brief intuitive introduction to topology, proximity, and nearness. It shows that one can explain the concept of continuity and its various modifications even to persons who are not mathematicians. Teachers of calculus know how difficult it is to teach continuity in the classroom; see for example, Devlin’s article on the website of the Mathematical Association of America: Will the real continuous function please stand up? (May 2000) It is not widely known that abstract pure mathematics has applications in daily life. There are many examples in mathematics but here we wish to explain topology and its applications. We use simple examples from day-to-day life to illustrate the concepts. Topology deals with the concept of nearness at various levels. Riesz first formulated this approach in an address to the International Congress of Mathematicians in Rome (1908). Level 1: Consider a typical family {mother, father, son, daughter}. We say that a person is near the family if that person is blood related to some member of the family. Of course, every member of the family is near the family and the family must first exist to talk about nearness! Grandparents, aunts, uncles, cousins… are persons near the family though they are not in the family. There are many other ways of defining such nearness relations, e.g. one may say that a person is near the family if the person helps the family in some way. In this definition the family physician, the plumber, the mailman,... are near the family. This concept is axiomatized with a few simple obvious conditions and one gets the abstract concept of a topological space. This was accomplished by the Polish mathematician K. Kuratowski in 1922. With a topological space, is associated another concept that of a continuous transformation. Suppose five years back P was a family physician of the family F. That is the Physician P was near the family F under the rule that P helps the family in some way. Today, after five *

Abstract of a public lecture given by the author in 2004 in the conference Subtle Technologies in Toronto, Canada.

2

Intuitive Introduction

years, both the Physician P and the family F have changed. If the Physician P is still near the family F, we say that it is a continuous relationship in day-to-day life and the same is said in mathematics. If for any reason, the Physician P ceases to be the family doctor of F, we say that the relationship is discontinuous. Thus a continuous transformation is one in which the nearness of a point to a set remains unchanged under that transformation. To recapitulate, in Topology, we have nearness relations between points and sets, together with continuous transformations which preserve these nearness relations. Level 2: At this level we talk about nearness between two families, technically called a proximity. This idea, already present in Riesz's work, was thoroughly studied by the Soviet mathematician V. Efremovič around 1940 and published in 1951. This idea was further developed by the Soviet mathematician Yu Smirnov. Again, this nearness between two families, can occur in several ways: (a) two families can be near because a daughter from one family has married a son from another, or (b) two families have a common friend (i.e., a person near both families), or (c) two families are interested in music and meet at a concert thus getting near each other. Moreover, in this subject one studies transformations which preserve nearness between pairs of sets. These are called proximally continuous transformations. To recapitulate, in Proximity, we have nearness relations between pairs of sets, together with proximally continuous transformations which preserve these nearness relations. Level 3: Here one talks of nearness of a number of families technically resulting in a generalization of a uniform space which was discovered by the French mathematician A. Weil in 1937. An example is that of the families of persons who work for the same company. These families get together for a picnic or a Christmas party. Again, one has transformations which preserve this nearness of families and these are called near transformations or uniformly continuous transformations. In the most general sense, this was first studied by the German mathematician Horst Herrlich in early 1970s. It is easy to see that the subject is international and mathematicians from all over the world have worked on this topic. Mathematicians work on this topic because the problems are interesting, challenging, or beautiful! Sometimes problems come from other areas but there are many instances where applications were discovered later. Abstract topology has found applications in Theoretical Computing, Quantum Mechanics, General Relativity, Mathematical Economics, Optimization, Convex Analysis, Probability Theory, Theory of Capacities, Child Psychology, a Model of our eyes, etc.

1

The Lodato Proximity “The existence of analogies between the central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to these central features.” - E. H. Moore

A topological space can be defined in terms of the Kuratowski closure axioms (1922) [Ke]. If the axioms are written in the form of “a point is near a set”, rather than in terms of the closure operator then they provide a motivation for the axioms for a Lodato or LO-proximity, which deals with nearness of two sets. This was done by Leader [Le 5] for the non-symmetric proximities and by his student Lodato [Lo 2] for the symmetric ones in 1964. LO-proximity is a generalization of intersection of two sets, which is replaced by nearness of two sets. LO-proximity and the earlier proximity discovered by Efremovič are finer structures than topology and provide a simple conceptual approach to many significant topological problems. In this chapter analogues of ultrafilters are introduced in a LO-proximity space and a simple T1 compactification is constructed [GN 1] which generalizes all known T1 and Hausdorff compactifications. It concludes with a study of generalized uniformities discovered by Mozzochi in 1968 [Moz 2]. MSC 2000: 54E05, 54D35, 54E15, 54E25.

Since ancient times, mathematicians have struggled with the precise formulation of continuity. An abstract, yet a simple formulation, was given by F. Riesz in 1908 [Ri], using the concept of nearness. Nearness is one of the rare concepts in the whole of mathematics that is at once intuitive and which can be made rigorous with little effort. Since one uses the words near and far in dayto-day life, the concept of nearness is, even for non-mathematicians, easy to understand. It can be explained to “the first person one meets in the street”, as the great mathematician, Joseph Louis Lagrange said. In addition, it can be made rigorous by formulating precise axioms. Its simplicity and depth provide a powerful tool in understanding many concepts as well as conducting research in Topology and Analysis. It is the principal purpose of this book to demonstrate this fact.

4

1 The Lodato Proximity

Thus Topology is a subject in which, given a nonempty set X, one studies a nearness relation which determines whether or not a point x in X is near a subset B of X. If there is another nonempty set Y with a similar nearness relation, then a function f : X Y is continuous if it preserves the nearness of points to subsets. As explained earlier, this can be easily related to the use of the term “continuous” in daily life and so one can explain continuity even to nonmathematicians. The category TOP consists of topological spaces as objects and continuous functions as morphisms. Other possibilities include nearness of 1. pairs of subsets (PROXIMITY) [Ef 1-2]. 2. finitely many subsets (CONTIGUITY) [II], [Iv]. 3. an arbitrary family of subsets (NEARNESS). [He 2], [Nai 4]. When there are two spaces with structures of the same kind, functions preserving proximity (res. contiguity, nearness) are called proximally continuous (res. contigual, near) maps. Obviously we get the categories PROX, CONT and NEAR. For more information on categorical implications see [He 2] and Section (4.5). Topology, proximity, contiguity, and nearness give progressively finer structures. As per Einstein’s quotation given above, it will be shown in this book that finer structures are useful in dealing with finer topological problems. Thus, the problem of finding necessary and sufficient conditions for the existence of continuous extensions of continuous functions from dense subspaces of Tychonoff spaces is purely topological. The problem has a simple solution in terms of proximity. Taimanov first proved a special case in which the range space is compact Hausdorff. His proof does not use proximity. However, even a glance at the statement of his result reveals the LO-proximity hidden in the domain and the (Efremovič) proximity hidden in the range! If the spaces involved are T1, then one needs contiguity.

1.1

Introduction

After an intuitive motivation given above, it is time for precise formulations. Let X be a nonempty set and let “x δ B” denote “x ∈ cl B” (where cl B = the closure of B) or “the point x is near the set B”. The negation “the point x is far from the set B” is denoted by “x δ B”. The well-known Kuratowski closure axioms are usually written in the following form: Let X be a nonempty set and let B, C be arbitrary subsets of X. A closure operator cl on X is a self-map on the power set of X which satisfies the following: (a) cl ∅ = ∅; (b) B ⊂ cl B; (c) cl (B ∪ C) = cl B ∪ cl C; (d) cl (cl B) = cl B;

1.1 Introduction

5

One can rewrite the above axioms using the term near. It is easy to see that the use of the term near in describing the closure axioms is logically equivalent to the above formulation. However, the following formulation is capable of generalization to proximity. Frequently, { x } is written as x. (1.1) Definition: Kuratowski closure axioms using “near”. Let X be a nonempty set. For each x ∈ X and B ⊂ X, C ⊂ X, (K1) (K2) (K3) (K4)

x δ B ⇒ B ≠ ∅, { x } ∩ B ≠ ∅ ⇒ x δ B, x δ (B ∪ C) ⇔ x δ B or x δ C, x δ B and b δ C for each b ∈ B ⇒ x δ C.

A topological space (X, T) is a set together with a Kuratowski closure operator cl or δ. As is well known, all other topological concepts such as open, closed, compact, continuous etc. can be expressed in terms of the closure operator “cl” given by cl B = { x ∈ X : x δ B }. For example, a subset E of X is called open provided that E = ∅ or each point of E is far from the complement of E i.e. for each x ∈ E, x δ E c. Traditionally, the topology T denotes the family of all open subsets of X [Ke]. A continuous function is one that preserves nearness of points to subsets. Formally, if X and Y have Kuratowski closure operators δ, δ’ respectively, then a function f : X Y is continuous iff for each x ∈ X and B ⊂ X, x δ B in X ⇒ f (x) δ’ f (B) in Y. From the categorical point of view, a morphism is a structure preserving map. In General Topology texts, a continuous function is defined as one for which the inverse image of any open set in the range is open in the domain [Ke]. This definition does not specify which structure is preserved by a continuous function. The definition given in the texts goes in the opposite direction! The Riesz approach is clear that (a) the topological structure consists of “nearness of points to sets” and (b) the continuous functions are those that preserve this structure. Next step is proximity. A proximity δ on a nonempty set X is concerned with nearness between two sets. There are many proximities in the literature, some symmetric and some nonsymmetric. One of the simplest proximities is called Lodato or LO-proximity. Leader studied the non-symmetric case [Le 5] and his student Lodato studied the symmetric one in 1964 [Lo 2]. The purpose of this book is to study symmetric proximities. The symmetric case satisfies the conditions (P0)−(P4) given below. A motivation for the axioms below is provided by the fact that if one replaces {x} by A in the Kuratowski closure operator (K1)−(K4), one gets (P1)−(P4). The symmetry axiom (P0) is added which is not relevant in the Kuratowski closure axioms.

6

1 The Lodato Proximity

Notation: A δ B means “A is near B” and its negation A δ B means “A is far from B” or “A is not near B”. (1.2) Definitions: Let X be a nonempty set. A Lodato proximity δ is one that satisfies the conditions (P0)−(P4) given below. For all subsets A, B, C of X (P0)

A δ B ⇒ B δ A, (symmetry)

(P1)

A δ B ⇒ A ≠ ∅ and B ≠ ∅,

(P2)

A ∩ B ≠ ∅ ⇒ A δ B,

(P3)

A δ (B ∪ C) ⇔ A δ B or A δ C.(union axiom)

(P4)

A δ B and { b } δ C for each b ∈ B ⇒ A δ C. (The LO-axiom)

Further δ is separated if it satisfies: (P5)

{ x } δ { y } ⇒ x = y.

A basic proximity δ is one that satisfies (P0)−(P3). As explained above, a motivation for the above axioms is provided by the fact that if one replaces A by {x} in (P1)−(P4) one gets precisely the Kuratowski closure axioms (K1)−(K4). Hence every Lodato proximity space (X, δ) has an associated topology T = T (δ); in this case the topology T and the LO-proximity δ are said to be compatible. In view of the symmetry condition (P0), the topology T (δ) induced by an LO-proximity δ satisfies the separation axiom R0 defined below. (1.3) The R0 axiom: x ∈ cl { y } ⇔ y ∈ cl { x }. A question arises if an R0 topological space has a compatible LO-proximity. The answer is yes. (1.4) Theorem: Every R0 topological space (X, T) has a compatible LO-proximity δ0 given by: A δ0 B ⇔ cl A ∩ cl B ≠ ∅. (Fine LO-proximity δ0) Proof: It is easily verified that δ0 satisfies (1.2) (P0)-(P3). (P4) Suppose A δ0 B and {b} δ0 C for each b ∈ B  Then cl A ∩ cl B ≠ ∅ and for each b ∈ B there is an x ∈ X such that x ∈ cl { b } ∩ cl C. Since X is R0, b ∈ cl { x } ⊂ cl C. So B ⊂ cl C and hence cl B ⊂ cl C. This implies cl A ∩ cl C ≠ ∅ and A δ0 C.

1.1 Introduction

7

Compatibility follows from (P4): x δ0 B ⇔ cl { x } ∩ cl B ≠ ∅. Hence there is a y ∈ cl { x } ∩ cl B. Since X is R0 , x ∈ cl { y } ⊂ cl B. Conversely, x ∈ cl B ⇒ cl { x } ⊂ cl B ⇒ cl { x } ∩ cl B ≠ ∅ ⇒ x δ0 B.

R0 spaces are also known as symmetric or weakly regular spaces. R0 is independent of T0. Both R0 and T0 are weaker separation axioms than T1. In fact, T1 = R0 + T0. In R0 spaces one cannot distinguish one point from other points which are in its closure. This creates a difficulty in embedding one space into another. So in this book, for the most part, all topological spaces are assumed to satisfy at least the separation axiom T1. (1.5) Remarks: Notation: B c = complement of B, int B = interior of B. (a) If δ is a LO-proximity, A δ B ⇔ cl A δ cl B. Proof: By the union axiom, A δ B, A ⊂ C, B ⊂ D ⇒ C δ D; hence A δ B ⇒ cl A δ cl B. On the other hand, x ∈ cl B ⇒ x δ B. So by the LO-axiom cl A δ cl B, and each element x of cl B is near B ⇒ cl A δ B. By symmetry, B δ cl A and repeating the argument, B δ A. Finally, by symmetry A δ B. (b) A δ B ⇒ (i) cl A ⊂ B c and (ii) A ⊂ int (B c). Proof: (i) A δ B ⇒ cl A δ B ⇒ cl A ⊂ B c. (ii) A δ B ⇒ A δ cl B ⇒ A ⊂ (cl B) c = int (B c). (c) If (X, δ) is a LO-proximity space and Y ⊂ X, one may define for subsets A, B of Y, A δ’ B in Y ⇔ A δ B in X. It is easy to check that δ’ is a LO-proximity on Y which is called the subspace LO-proximity induced by δ. It is easy to check that if δ is a LO-proximity compatible with the topology T on X then the subspace LO-proximity δ’ is compatible with the subspace topology induced by T on Y. (d) It will be shown later that the fine LO-proximity δ0 is the most important proximity. This is so because every LO-proximity, abstractly defined on a set X, is the subspace LOproximity induced by the fine LO-proximity δ0 on a suitable compactification of X (1.21). (e) Partial Order: LO-proximities are partially ordered as follows: δ > δ’

⇔ for all subsets A, B of X, A δ B ⇒ A δ’B; ⇔ for all subsets C, D of X, C δ’ D ⇒ C δ D.

We also write δ’ < δ for δ > δ’. If δ > δ’, then δ is said to be finer than δ’ or δ’ is coarser than δ. In this partial order, δ0 is the finest compatible LO-proximity on an R0 topological

8

1 The Lodato Proximity

space X. This is so because if δ is any compatible LO-proximity on X, A δ0 B ⇔ cl A ∩ cl B ≠ ∅ ⇒ cl A δ cl B ⇒ A δ B. So δ0 > δ. (f) A topology T on X is T1 ⇔ every compatible LO-proximity δ on X is separated that is any two distinct points are far. (g) The coarsest compatible LO-proximity on a T1-topological space (X,T) is given by: (1.6)

A δC B ⇔ cl A ∩ cl B ≠ ∅ or both A and B are infinite.

Proof: It is easy to see that δC satisfies (P.0)–(P.2). The union axiom is satisfied because of cl (B ∪ C) = cl B ∪ cl C and the union of two sets is infinite iff at least one of them is so. The LO-axiom: Suppose A δC B and { b } δC C for each b ∈ B. A δC B ⇒ cl A ∩ cl B ≠ ∅ or A, B are both infinite. For each b ∈ B, { b } δC C ⇒ B ⊂ cl C ⇒ cl B ⊂ cl C. If cl A ∩ cl B ≠ ∅, then cl A ∩ cl C ≠ ∅ ⇒ A δC C. If A, B are infinite, then A, C are infinite ⇒ A δC C. Compatibility: { x } δC E ⇔ x ∈ cl E, since { x } is finite. If δ is any compatible LOproximity, A δ C B ⇒ cl A ∩ cl B = ∅ and cl A or cl B is finite. A finite set is far from a disjoint closed set in any compatible LO-proximity and so A δ B. Hence δ > δC; so δC is the coarsest compatible LO-proximity. This strange looking LO-proximity δC, which we call the coarse LO-proximity, is useful in the construction of counter examples. (h) A cousin of the coarse LO-proximity is the LO-proximity δ on complex numbers C or real numbers R defined by A δ B ⇔ cl A ∩ cl B ≠ ∅ or both A and B are uncountable. (1.7) Examples: Here we give examples of some important LO-proximities. Some of them satisfy stronger properties and have distinguishing names as will be seen in the sequel. We have already seen two above viz. the fine and coarse LO-proximities. There are two trivial LO-proximities, which are useful in the construction of counter examples: (a) A α B ⇔ A ∩ B ≠ ∅ which takes us back to Set Theory! (Discrete) (b) A β B ⇔ A, B are nonempty. (Indiscrete) How to construct compatible LO-proximities on a given T1 space or a Tychonoff space or a space with a (semi-) metric? One may define either the nearness of two sets or their farness. To define nearness of two sets one sufficient, but not a necessary condition is that their closures intersect. To allow nearness of two sets whose closures are disjoint, one has to add an alternate condition that satisfies the union axiom such as the sets are infinite or their closures are non-compact (1.9), (1.5); (g) & (h).

1.1 Introduction

9

To define farness of two sets one may use the functional separation by continuous or upper semi-continuous functions (1.11) or use the condition that they are at a positive distance apart (1.8). (a) In a metric space (X, d) define the gap between two sets A, B: d (A, B) = inf { d (a, b): a ∈ A and b ∈ B }, = ∞ if A or B is empty. The metric d induces the metric proximity δm. (1.8) Metric proximity: A δm B ⇔ d (A, B) = 0. Proof: Here we show that the metric proximity is LO; later we will see that it satisfies a stronger condition. Obviously δm satisfies (P0)-(P2). (P3) d (A, B ∪ C) = min { d (A, B), d (A, C) }. So A δm (B ∪ C) ⇔ d (A, B ∪ C) = 0 ⇔ d (A, B) = 0 or d (A, C) = 0 ⇔ A δm B or A δm C . (P4) A δm B and { b } δm C for each b ∈ B ⇒ d (A, B) = 0 and d(b,C) = 0 for each b∈B. By triangle inequality, d(A,C) = 0 ⇒ Aδm C.

(b) In Complex Analysis it is customary to embed C, the set of complex numbers which form a non-compact space, into the unit sphere which forms a compact space and consider the north pole as the point at infinity ∞. In this setting ∞ is near all subsets of C whose closures are non-compact, i.e. any two unbounded sets are near. This construction was generalized by P.S. Alexandroff ([Ke] Page 150) to get the one-point compactification of any non-compact T1 space (X, T). Let ∞ be an element not in X. Set X* = X ∪ { ∞ }. Let the neighbourhoods of points x ∈ X be the same as in T and let the neighbourhoods of ∞ be the complements of closed compact subsets of X. Then X* is the T1 Alexandroff one-point compactification of X. If X is compact and we follow the above construction, then ∞ is an isolated point of X*.

In the case of a locally compact Hausdorff space, this construction yields a Hausdorff compactification as will be seen in the next chapter (2.3) (b). (1.9) Alexandroff LO-proximity δA: In a T1 space (X, T). A δA B ⇔ A δ0 B or both cl A and cl B are non-compact, is a compatible LO-proximity on X. Proof: Obviously δA satisfies (P0)-(P2).

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1 The Lodato Proximity

(P3) The union axiom is satisfied since the closure of the union of two sets is non-compact if and only if at least the closure of one of them is non-compact. (P4) Suppose A δA B and { b } δA C for each b ∈ B. If A δ0 B we are done. If not then both cl A and cl B are non-compact. Since b δA C for each b ∈ B, it follows that cl B ⊂ cl C, cl C is non-compact and A δA C. Compatibility follows from the fact that x δA E ⇔ x ∈ cl E , since { x } is compact.

(c) R, the set of real numbers with the usual topology is a non-compact space. In topological measure theory, R ∪ { −∞, ∞ }, the two-points compactification of R, plays a significant role. (1.10) The Two-points-proximity δT: It will be seen later that it satisfies Efremovič condition stronger than LO. A δT B ⇔ A δ0 B or [inf A = inf B = − ∞] or [sup A = sup B = ∞]. - ∞ -------------------------------------------------------------------------------------------------- + ∞ R Proof: Obviously δT satisfies (P0)-(P2). (P3) The union axiom is satisfied because union of two sets is unbounded above (respectively, below) iff at least one is so. (P4) and the compatibility follow from the fact x δT E ⇔ x ∈ cl E. It is easy to see that the two-points-proximity δT is induced by the two-points-compactification R ∪ {− ∞, ∞ } of R.

(d) In a Tychonoff space (X, T) we have (1.11) the Functionally-separating-proximity δF : A δ F B ⇔ there is a continuous function f : X

[0, 1] with

f (A) = 0 and f (B) = 1. For a proof see (2.3) (c) where we show that it satisfies Efremovič condition stronger than the LO-axiom. We will show later that δF is induced by the Stone-Čech compactification βX of X (2.14) (b).

(e) In a uniform space (X, U) the Uniformity-induced proximity δ = δ (U) is defined by: A δ B ⇔ there is a U ∈ U such that U (A) ∩ B = ∅.

1.1 Introduction

11

For a proof that δ is a proximity see (1.25); in the next chapter we show that it is a stronger Efremovič proximity.

(f) Topological Group: Let (X, · , T) be a topological group and let N be the neighbourhood system of the identity. For U ∈ N, A ⊂ X set UA = { u · a : u ∈ U and a ∈ A}, AU = { a · u : u ∈ U and a ∈ A}, We may define two LO-proximities on X as follows: (i) A δ B ⇔ for each U ∈ N, UA ∩ B ≠ ∅. (ii) A δ B ⇔ for each U ∈ N, AU ∩ B ≠ ∅. In general, the two LO-proximities differ and satisfy Efremovič condition stronger than LO. They are equal if X is commutative or compact. We note here that the concept of nonempty intersection of finite sets is useful in Topology, as for example, its use in filters. When the concept of nonempty intersection is generalized to nearness of two or more sets one has generalizations of a filter. This gives access to many more subtle cases as the above examples show. In fact, on the space of real numbers, there are uncountably many different proximities [Sh 2]. Herein lies the secret of success of the concept of proximity in solving many topological problems. (1.12) Definition: Proximal neighbourhood. In a LO-proximity space (X, δ), let A, B be subsets of X and Β c = X−B. Then B is called a proximal neighbourhood of A iff A δ Β c and is written A