*308*
*30*
*882KB*

*English*
*Pages 211*
*Year 1996*

- Author / Uploaded
- Banakh T.
- Radul T.
- Zarichnyi M

T. Banakh, T. Radul, M. Zarichnyi

Absorbing sets in Infinite-Dimensional Manifolds

Mathematical Studies Monograph Series Volume 1

VNTL Publishers Lviv, 1996

2

Taras Banakh, Lviv State University Taras Radul, Institute for Applied Problems of Mechanics and Mathematics Michael Zarichnyi, Lviv State University

3

Preface

Many remarkable topological spaces can be characterized by means of short list of their topological properties. In early 70es the famous characterization theorems for Hilbert space manifolds and Hilbert cube manifolds were proved by Henryk Toru´ nczyk. His technique was essentially based on the completeness of the considered spaces. A program of proving noncomplete counterparts of the Toru´ nczyk theorems was realized in the fundamental paper of M.Bestvina and J.Mogilski. Starting from Anderson’s cap-sets, Bessaga-Pelczy´ nski’s skeletoids, and absorbing sets of J.West they cristallized the notion of absorbing set in Hilbert space manifold. These absorbing sets form an important class of spaces for which characterization theorems can be proved. Besides, they possess many remarkable topological properties, in particular, their topological and homotopy classiﬁcations coincide. The book is devoted to the theory of absorbing sets and its applications. It is divided into ﬁve chapters. The ﬁrst chapter provides an exposition of the basic theory of absorbing sets. Our approach diﬀers from that of Bestvina and Mogilski and is based on the ﬁrst author’s characterization of spaces that can be homotopy densely embedded into Hilbert space manifolds. This allows us to simplify substantially our exposition and, moreover, to show the equivalence of the relative (Bestvina, Mogilski) and absolute (Dobrowolski, Mogilski) notions of absorbing sets. The term “absorbing space” is used for the latter situation and we prefer it in our book (the title – to be more recognizible – is an exception!). Bestvina and Mogilski constructed absorbinhg sets in the absolute Borel classes. In the second chapter we give examples of absorbing sets for classes of spaces deﬁned not only by set-theoretical but also by dimensional conditions. The third chapter is devoted to some advanced topics in the theory. In particular, here we examine the mutual relation between absorbing spaces and pairs. Absorbing spaces have found various applications in topology and other disciplines. We present some of them in the fourth chapter (inﬁnite products, hyperspaces, topological groups) and in the ﬁfth chapter (convexity and related topics). Lviv, July 1996

4

Contents1 Preface Chapter I: Basic Theory 1.1. Preliminaries 1.2. Homotopy dense and homotopy negligible sets 1.3. The strong discrete approximation property 1.4. Z-Sets and strong Z-sets 1.5. Strong universality 1.6. Absorbing and coabsorbing spaces 1.7. Strongly universal and absorbing pairs

3 6 5 13 17 23 29 38 44

Chapter II: Constructions of absorbing spaces 2.1. Preliminaries I: Descriptive Set Theory 2.2. Preliminaries II: Dimension Theory 2.3. Invertible and soft maps 2.4. Absorbing spaces for [0, 1]-stable classes 2.5. Weak inverse limits and absorbing sets 2.6. Some negative results

53 53 56 69 81 86 92

Chapter III: Advanced Topics 3.1. Interplay between strongly universal spaces and pairs 3.2. Characterizing (strong) C-universality for “nice” classes C

95 95 106

Chapter IV: Applications I. 4.1. Inﬁnite products 4.2. Topological groups 4.3. Hyperspaces

125 125 128 137

Chapter V: Applications II: Convex Sets 5.1. Locally compact convex sets 5.2. Topologically complete convex sets 5.3. Strong universality in convex sets 5.4. Strong universality in locally convex spaces 5.5. Some counterexamples 5.6. Spaces of probability measures 5.7. Function spaces Cp (X) and Cp∗ (X)

147 147 150 154 166 172 188 196

References

199

1 The

numbering does not coincide with the printed version.

1.1. PRELIMINARIES

5

Chapter I Basic Theory

The main results of this Chapter are Uniqueness Theorems 1.6.3, 1.6.4 on topological equivalence of (co)absorbing spaces. We develop all the necessary material in Sections 1.1—1.5. Section 1.7 contains Uniqueness Theorem 1.7.6 for absorbing pairs and Theorem 1.7.9 revealing interplay between strongly universal spaces and pairs. §1.1. Preliminaries. A. General conventions and notation. By a space we mean a separable metrizable topological space; a compactum, by deﬁnition, is a compact space. Observe that each space has a countable base, is Lindel¨of and paracompact. Unless stated to the contrary, all maps and functions between spaces are assumed to be continuous. We let I denote the interval [0, 1], I k the k-dimensional cube, and ∂I k its boundary. The space of all real numbers is denoted by R and N denotes the set of all natural numbers. Sometimes, N will be considered also as a discrete topological space. The Hilbert cube Q is the countable inﬁnite product Q = [−1, 1]ω . We will consider also the following subspaces of Q: the pseudo-interior s = (−1, 1)ω , the pseudo-boundary Bd(Q) = Q\s, the radial interior Σ = {(ti ) ∈ Q | supi |ti | < 1} of the Hilbert cube, and subspaces Qf = {(ti ) ∈ Q | ti ̸= 0 for ﬁnitely many of i} and σ = Qf ∩ s. The standard separable Hilbert space is denoted by l2 and lf2 = {(ti ) ∈ l2 | ti ̸= 0 for ﬁnitely many of i}. For any space X we let d denote an admissible metric on X, i.e. a metric that generates the topology. In the sequel we consider only admissible

6

BASIC THEORY

metrics. Many of spaces we deal with are concrete objects and then d will be deﬁned explicitly. If A ⊂ X then diam(A) denotes the diameter of A. Similarly, A¯ or ClX (A) denotes the closure of A and Int(A) denotes its interior. As usual, ¯ Int(A). Finally, B(A, ε) = Bd(A) is the boundary of A, i.e. Bd(A) = A\ {y ∈ X | d(A, y) < ε} and D(A, ε) = {y ∈ X | d(A, y) ≤ ε} denote the open and closed balls about A with radius ε, respectively. If A = {x} for some x then we write B(x, ε) instead of B({x}, ε) and D(x, ε) instead of D({x}, ε), respectively. If f : X → Y is a function and A ⊂ X then f |A denotes the restriction of f onto A. For a space X by id : X → X (or, sometimes, by idX ) we denote the identity map. A metric d for X is called bounded if diam(X) < ∞. Noticing that min{1, d} is also an admissible metric for X, we see that each metric can be replaced by one which is bounded. The symbol ∼ = means homeomorphic to. A Polish space is a complete-metrizable separable topological space. A Baire space, by deﬁnition, is a space X such that for every countable ∩ collection {Gn }n∈N consisting of open dense subsets of X, the intersection n∈N Gn is dense in X. By M0 , M1 , and A1 we denote the classes of compact, Polish and σ-compact spaces respectively. For a subset A of a linear space L by conv(A), span(A), and aﬀ(A) we denote respectively the convex, linear, and aﬃne hull of A in L. For subsets F ⊂ R, A, B ⊂ L we let F · A = {t · a | t ∈ F, a ∈ A} and A + B = {a + b | a ∈ A, b ∈ B}. It will be convenient to introduce the following notation. For U, V collections of subsets of a space X, the symbol U ≺ V means that U is inscribed into V (or U reﬁnes V), i.e. for every U ∈ U there is V ∈ V with U ⊂ V . Let X be a space, A ⊂ X, and U a collection of subsets of X. Let U ∩ A = {U ∩ A | U ∈ U}. The star of A with respect to U is the set St (A, U) = ∪{U ∈ U | A ∩ U ̸= ∅}. The collection {St (U, U) | U ∈ U} is denoted by St (U). For every n ≥ 0 deﬁne St n (U) inductively letting St 0 (U) = U and St n (U) = St (St n−1 (U)). We set mesh U = sup{diam(U ) | U ∈ U}. For a map f : Y → X let f −1 U = {f −1 (U ) | U ∈ U}. Recall that a collection U of subsets of X is called locally ﬁnite in X (resp. discrete in X) if each point x ∈ X has a neighborhood that meets only ﬁnitely many of elements of U (resp. at most one element of U). Let ε > 0. We say that two maps f, g : Y → X are ε-close if d(f (y), g(y)) < ε for every y ∈ Y . This notion can be generalized as follows.

1.1. PRELIMINARIES

7

Suppose U is a collection of subsets of X. We say that two maps f, g : Y → X are U-close and denote this by (f, g) ≺ U if for every y ∈ Y with f (y) ̸= g(y) there is U ∈ U such that f (y), g(y) ∈ U . Sometimes this notion will be used in a more general situation: given two maps f : Y → X and f ′ : Y ′ → X (with distinct domains) we write (f, f ′ ) ≺ U if the restrictions f |Y ∩ Y ′ and f ′ |Y ∩ Y ′ are U-close. By cov(X) we denote the set of all open covers of X. In the sequel, we will say that a map f : Y → X can be approximated by maps with certain property if for every cover U ∈ cov(X) there exists a map g : Y → X which possesses that property and is U-close to f . For spaces Y, X by C(Y, X) we denote the space of all functions Y → X, endowed with the limitation topology whose neighborhood base at an f ∈ C(Y, X) consists of the sets B(f, U) = {g ∈ C(Y, X) | (g, f ) ≺ U }, where U runs over the set cov(X). If Y is compact then the limitation topology on C(Y, X) coincides with the compact-open topology, and consequently is metrizable and separable. A homotopy h : Y × [0, 1] → X is called a U-homotopy if {h({y} × [0, 1])}y∈Y ≺ U; h is deﬁned to be an ε-homotopy, where ε > 0, if diam h({y} × [0, 1]) < ε for every y ∈ Y . Two maps f, g : Y → X are called U-homotopic (resp. ε-homotopic) if they can be linked by a U-homotopy (resp. by an ε-homotopy). As usual, for a homotopy h : Y × [0, 1] → X and t ∈ [0, 1] we let ht : Y → X be the map deﬁned by ht (y) = h(y, t). Recall that a perfect map is a map f : Y → X such that the preimage f −1 (K) of any compact set K ⊂ X is compact. It is well known that a map is perfect if and only if it is closed and the preimage of any point is compact. For a cover U ∈ cov(X), a map f : X → Y is deﬁned to be a U-map, provided f −1 V ≺ U for some cover V ∈ cov(Y ). Under a partition of unity on X we understand any collection of functions {λi : X → [0,∑ 1]}i∈I such that 1) the family {λ−1 i (0, 1]}i∈I is locally ﬁnite in X, and 2) i∈I λi (x) = 1 for every x ∈ X. We will use quite often the following fact: for every open cover U of X there exists a partition of unity {λU : X → [0, 1]}U ∈U such that λ−1 U (0, 1] ⊂ U for every U ∈ U. Finally, to avoid misunderstanding, we impose the following preference order on set-theoretic operations: \, ×, ∩, ∪ (i.e., A × B\C means A × (B\C)). “Iﬀ” is an abbreviation for “if and only if”. Exercises to §1.1.A. 1. Show that for any open cover U of a subspace X ⊂ Y there is a cover U˜ of some neighborhood U ⊂ Y of X such that U˜ ∩ X = U .

We shall say that a function f : X → R is Lipschitz if |f (x) − f (y)| < d(x, y) for every x, y ∈ X. 2. Show that for every open cover U ∈ cov(X) there exists a Lipschitz function ε : X → (0, 1] such that {B(x, ε(x)) | x ∈ X} ≺ U . Hint: Using paracompactness of X, ﬁnd a locally ﬁnite cover V ∈ cov(X), inscribed into U and consider the functions

8

BASIC THEORY ε′ (x) = max{d(x, X\V ) | V ∈ V} and ε(x) = min{1, ε′ (x)}.

3. Suppose F is a locally ﬁnite (discrete) collection of subsets of a space X. Find a cover U ∈ cov(X) such that the collection {St (F, U ) | F ∈ F} is still locally ﬁnite (discrete). Hint: For every point x ∈ X pick a neighborhood Vx of x that meets only ﬁnitely many of elements of F (at most one element of F), and let U ∈ cov(X) be any cover with St U ≺ {Vx | x ∈ X}. 4. Let X be a space and U ⊂ X × [0, 1] a neighborhood of X × {0}. Find a function ε : X → (0, 1] such that {(x, t) | 0 ≤ t ≤ ε(x)} ⊂ U . Hint: For every x ∈ X ﬁx a neighborhood Vx ⊂ X and a real εx ∈ (0, 1] such that Vx × [0, εx ] ⊂ U . Using paracompactness of X ﬁnd a partition of unity on X {λx : X → [0, 1]}x∈X such that λ−1 1] ⊂ Vx for every x. Deﬁne ﬁnally the required map ε : X → (0, 1] letting x (0, ∑ ε(z) = λ (z)εx for z ∈ X. x∈X x 5. Let F be a collection of subsets of a space X. Prove the following statements: a) F is discrete if and only if F is locally ﬁnite and F¯ ∩ F¯ ′ = ∅ for distinct F, F ′ ∈ F ; b) F is locally ﬁnite if and only if the collection {F¯ | F ∈ F} is locally ﬁnite; c) If X is a subspace in Y and F is a locally ﬁnite collection in X then F is locally ﬁnite in some open neighborhood U ⊂ Y of X. 6. Show that any locally ﬁnite open cover of a compact space is ﬁnite. 7. Show that for every perfect map f : K → X of a locally compact space K there exists a cover U ∈ ∪ cov(X) such that every map g : K → X U -close to f is perfect. Hint: ∞ Write K = n=1 Kn , where each Kn ⊂ Int Kn+1 is compact and notice that the collection {f (K\ Int(Kn ))}n≥1 is locally ﬁnite in X. Using exercise 3, ﬁnd a cover U ∈ cov(X) such that the collection {St (f (K\ Int(Kn )), U )}n≥1 is still locally ﬁnite. The cover U is required. 8. (Convergence Principle) Let {Un }n≥0 be a sequence of covers of a space X such that St Un ≺ Un−1 and mesh Un < 2−n for every n ≥ 1, and let {fn : Y → X}n≥0 be a sequence of maps such that (fn , fn−1 ) ≺ Un for each n ≥ 1. a) Assuming that for every y ∈ Y the sequence (fn (y)) has a pointwise limit f (y), ¯ | U ∈ U0 }; show that the function f : Y → X is continuous and (f, f0 ) ≺ U¯0 = {U b) Assuming that the metric d on X is complete, show that for every y ∈ Y the sequence (fn (y)) is convergent. Hint: Notice that the sequence (fn ) is uniformly Cauchy. 9. Suppose f : K → X is an injective map and {Ki }i∈I is a closed cover of K such that the collection {f (Ki )}i∈I is locally ﬁnite in X and for every i ∈ I the restriction f |Ki is a closed embedding. Show that f is a closed embedding. 10. Let f : K → M be an embedding and C ⊂ K, X ⊂ M subsets such that f (C) ⊂ X. Suppose C is dense in K and f (C) is closed in X. Show that C = f −1 (X). 11. Prove the following statements concerning the space C(Y, X). a) The limitation topology on C(Y, X) coincides with the topology whose neighborhood base at an f ∈ C(Y, X) consists of the sets B(f, ε) = {g ∈ C(Y, X) | d(g(y), f (y)) < ε(f (y)) for every y ∈ Y }, where ε runs over the set C(X, (0, ∞)). b) If Y is compact then the limitation topology on C(Y, X) coincides with the compactopen topology, and consequently is separable and metrizable. c) If Y is compact and X is Polish then the space C(Y, X) is Polish. d) If X is compact then the space C(Y, X) is metrizable but in general, not separable. e) The sets B(f, U ), in general, are not open (but have f as an interior point). f) If X is a Polish space then every closed subspace F ⊂ C(Y, X) is a Baire space.

BASIC FACTS FROM ANR-THEORY

9

g) If Y is locally compact then the set of perfect maps is open in C(Y, X). h) If F is a collection of subsets of Y then the sets {f ∈ C(Y, X) | f (F) is a locally ﬁnite collection in X} and {f ∈ C(Y, X) | f (F) is a discrete collection in X} are open in C(Y, X). i) For every U ∈ cov(Y ) the set of all U -maps is open in C(Y, X). j) If Y is a Polish space then the set of all closed embedding is a Gδ -set in C(Y, X).

B. Basic facts from ANR-theory. We presume that the reader is familiar with the theory of absolute neighborhood retracts, so here we only list the results of this theory we will need in the sequel. A retraction is a map r : X → X with the property r ◦ r = r. A subset A ⊂ X is called a retract of X, provided A = r(X) for some retraction r : X → X. A space X is deﬁned to be an absolute retract (brieﬂy an AR) if X is a retract in every space Y containing X as a closed subset. Correspondingly, X is called an absolute neighborhood retract (brieﬂy an ANR) if for every space Y containing X as a closed subspace, X is a retract of some open neighborhood U of X in Y . A space X is deﬁned to be an absolute (neighborhood) extensor (brieﬂy an A(N)E) provided for every space A, and a closed subspace B every map f : B → X can be extended to a map f¯ : A → X (to a map f : U → X of some neighborhood U ⊂ A of B). 1.1.1. Proposition. A space is an absolute (neighborhood) retract if and only if it is an absolute (neighborhood) extensor. This fact is a corollary of the following two results. 1.1.2. Any convex subspace of a locally convex linear metric space is an absolute extensor. 1.1.3. Any space can be embedded as a closed subset into a linear normed space. Let us recall the following properties of ANR’s. 1.1.4. Every open subspace of an ANR is an ANR as well. 1.1.5. A space X is an ANR if and only if every point x ∈ X has a neighborhood that is an ANR. 1.1.6. A space is an AR if and only if it is a contractible ANR. Recall that a space is called (strongly) countable-dimensional, provided it can be expressed as a countable union of its (closed) ﬁnite-dimensional subspaces.

10

BASIC THEORY

1.1.7. (Haver’s Theorem) A countable-dimensional space is an ANR if and only if it is locally contractible. A subset A ⊂ X is deﬁned to be contractible in X if there is a homotopy h : A × I → X such that h0 = idA and h1 (A) = {point}. 1.1.8. Each X ∈ ANR admits a cover consisting of open sets contractible in X. 1.1.9. For every map f : Y → X of a space Y into an X ∈ ANR and every covers U ∈ cov(Y ), V ∈ cov(X) there are a locally ﬁnite countable simplicial complex K, a surjective U-map p : Y → K, and a map q : K → X such that (q ◦ p, f ) ≺ V. 1.1.10. For every ANR X and every cover U ∈ cov(X) there exists a cover V ∈ cov(X) such that any two V-close maps into X are U-homotopic. 1.1.11. For every ANR X and every cover U ∈ cov(X) there exists a cover V ∈ cov(X) such that, for every map f : A → X, and every closed subset B ⊂ A, every map g : B → X, V-close to f |A, can be extended to a map g¯ : A → X, U-close to f . Propositions 1.1.1–1.1.11 are well known and can be found in any textbook on ANR-theory, see, e.g., K.Borsuk [1967], S.T. Hu [1965], or J.van Mill [1989]. The following statement is a little bit less known. 1.1.12. Let C be a subspace of a space K and let X be an ANR. For every map f : C → X and every cover V ∈ cov(X) there is a map g : W → X of a neighborhood W of C in K such that (g|C, f ) ≺ V. Proof. According to 1.1.3, X can be considered as a closed subspace of a normed space L. Fix a map f : C → X and a cover V ∈ cov(X). Since X is an ANR, there exists a retraction r : V → X of some open neighborhood V of X in L. Using continuity of r, for every x ∈ X pick an open convex neighborhood Vx of x in V such that (1)

{(Vx ∩ X) ∪ r(Vx ) | x ∈ X} ≺ V.

Let V ′ = ∪{Vx | x ∈ X} and let V ′ ∈ cov(V ′ ) be any cover with St V ′ ≺ {Vx | x ∈ X}. Pick a cover U ∈ cov(C) such that U ≺ f −1 V ′ . For every ˜ ⊂ K with U ˜ ∩ C = U . Let U ∈ U ﬁx a point cU ∈ U and an open set U ˜ | U ∈ U}, U˜ = {U ˜ | U ∈ U}, and let {λU : W → [0, 1]}U ∈U be a W = ∪{U ˜ partition of unity such that λ−1 U (0, 1] ⊂ U for every U ∈ U. Finally, deﬁne a map g : W → X by the formula g(z) = r

(∑ U ∈U

) λU (z)f (cU ) ,

z ∈ W.

SURVEY ON s-MANIFOLDS AND Q-MANIFOLDS

11

To show that (g|C, f ) ≺ V, ﬁx any c ∈ C, and consider the ﬁnite subset F = {U ∈ U | λU (c) ̸= 0}. Then for every U ∈ F f (cU ) ∈ St (f (c), V ′ ) ⊂ Vx for some x ∈ X. Since Vx is convex, we get {f (c), g(c)} ⊂ (Vx ∩ X) ∪ r(Vx ). This together with (1) immediately imply (g|C, f ) ≺ V. Remark that 1.1.12 is close in spirit to Lavrentiev Theorem, which will be used quite often in this book. 1.1.13. Theorem. Let X, Y be Polish spaces and A ⊂ X, B ⊂ Y subsets. Every map (homeomorphism) h : A → B can be extended to a map ˜ : A˜ → B ˜ between some Gδ -subsets A˜ ⊂ X and B ˜ ⊂Y. (homeomorphism) h Exercises to §1.1.B. 1. Prove that ﬁnite or countable product of AR’s is an AR. 2. Prove that for at most countable index set A, the product ANR iﬀ Xα are ARs for all but ﬁnitely many α.

∏ α∈A

Xα of ANRs is an

3. Let X be a compact AR. Prove that Cone(X) = (X × I)/(X × {0}) (the cone of X) is an AR.

C. Survey on s-manifolds and Q-manifolds. Let X be a space. A space M is deﬁned to be an X-manifold if each point x ∈ M has a neighborhood homeomorphic to an open subset of X. In this case we say that M is modeled on X, and X is called a model space. Below we list basic results of the theory of s-manifolds, i.e. manifolds modeled on the pseudo-interior s of the Hilbert cube. All the cited results can be found in the book of C.Bessaga, A.Pelczy´ nski [1975] and the fundamental paper by Toru´ nczyk [1981]. It should be noticed that the pseudointerior s is homeomorphic to the Hilbert space l2 (see, Anderson [1966] or J. van Mill [1989]), so all that will be said about s-manifolds concerns also l2 -manifolds. 1.1.14. (Characterization Theorem). A space X is an s-manifold if and only if a) X is a Polish ANR and b) for every cover U ∈ cov(X) and a map f : Q × N → X there is a map f¯ : Q×N → X such that (f¯, f ) ≺ U and the collection {f¯(Q×{n})}n∈N is discrete in X. 1.1.15. (Classification by Homotopy Type). Two s-manifolds are homeomorphic if and only if they are homotopy equivalent. 1.1.16. (Triangulation Theorem). Every s-manifold M is homeomorphic to the product K ×s for some locally ﬁnite countable simplicial complex K.

12

BASIC THEORY

1.1.17. (ANR-Theorem). For every Polish ANR X the product X × s is an s-manifold. A subset A ⊂ X is deﬁned to be a Z-set in X if A is closed and for every cover U ∈ cov(X) there is a map f : X → X such that (f, id) ≺ U and f (X) ∩ A = ∅. A subset A ⊂ X is called a Zσ -set in X if it can be written as a countable union of Z-sets of X. An embedding f : A → X is called a Z-embedding, provided its range f (A) is a Z-set in X. 1.1.18. (Z-Set Unknotting Theorem). Let M be an s-manifold and U ∈ cov(M ). Every homeomorphism h : A → B between two Z-sets A, B in ¯:M →M M , U-homotopic to id|A, can be extended to a homeomorphism h of whole M , U-homotopic to idM . 1.1.19. (Strong Negligibility Theorem). For every s-manifold M , every cover U ∈ cov(M ), and every Zσ -set A ⊂ M there exists a homeomorphism h : M → M \A which is U-close to id. 1.1.20. (Open Embedding Theorem). For s-manifolds M , N , a cover U ∈ cov(N ), and a map f : M → N there exists an open embedding f¯ : M → N which is U-close to f . 1.1.21. (Strong M1 -Universality). Let M be an s-manifold, U ∈ cov(M ) a cover, A a Polish space, B a closed subset of A, and f : A → M a map such that the restriction f |B : B → M is a Z-embedding. Then there exists a Z-embedding f¯ : A → M such that f¯|B = f |B and (f¯, f ) ≺ U . Moreover the map f¯ can be chosen in such a way that f¯(A\B)∩Z = ∅ for any pregiven Zσ -set Z in M . 1.1.22. (Perfect Retraction Theorem). Any perfect retract of an smanifold M is an s-manifold. We will also need some facts from the theory of Q-manifolds, i.e. manifolds modeled on the Hilbert cube. A good reference on Q-manifolds is the books of T.Chapman [1976], J. van Mill [1989], and the paper of Toru´ nczyk [1980]. 1.1.23. (Characterization Theorem). A space X is a Q-manifold (resp. X ∼ = Q) if and only if a) X is a locally compact ANR (reps. X is a compact AR) and b) for every cover U ∈ cov(X) and a map f : Q × {1, 2} → X there is a map f¯ : Q × {1, 2} → X such that (f¯, f ) ≺ U and f¯(Q × {1}) ∩ f¯(Q × {2}) = ∅. Sometimes, it is convenient to apply the following condition equivalent to b) b′ ) for every ε > 0, every n ∈ N, and every maps f1 , f2 : I n → X there are maps f1′ , f2′ : I n → X such that d(fi , fi′ ) < ε, i = 1, 2, and f1′ (I n ) ∩ f2′ (I n ) = ∅.

1.2. HOMOTOPY DENSE AND NEGLIGIBLE SETS

13

1.1.24. (ANR-Theorem). For every locally compact ANR X the product X × Q is a Q-manifold; and for every compact AR Y the product Y × Q is homeomorphic to Q. 1.1.25. (Z-Set Unknotting Theorem). Let M be a Q-manifold and U ∈ cov(M ). Every homeomorphism h : A → B between two Z-sets A, B in ¯:M →M M , U-homotopic to id|A, can be extended to a homeomorphism h of whole M , U-homotopic to id : M → M . 1.1.26. (Strong M0 -Universality). Let M be an Q-manifold, U ∈ cov(M ) a cover, A a compact space, B a closed subset of A, and f : A → M a map such that the restriction f |B : B → M is a Z-embedding. Then there exists a Z-embedding f¯ : A → M such that f¯|B = f |B and (f¯, f ) ≺ U. Moreover the map f¯ can be chosen in such a way that f¯(A\B) ∩ Z = ∅ for any pregiven Zσ -set Z in M . Finally, let us state the Anderson-Kadec Theorem. Recall that a Fr´echet space, by deﬁnition, is a locally convex linear complete metric space. 1.1.27. Every inﬁnite-dimensional Fr´echet space is homeomorphic to l2 . Exercises to §1.1.C. 1. Show that a locally compact ANR X is a Q-manifold if and only if the diagonal ∆X = {(x, x) | x ∈ X} is a Z-set in X × X. 2. Use Characterization Theorem in order to show that Cone(Q) = (Q × I)/(Q × {0}) is homeomorphic to Q. Hint: see T.Chapman [1976]. ∞ 3. Let Prove that ∏∞ {Xi }i=1 be a sequence of compact nondegenerate ARs. X is homeomorphic to Q. i=1 i 4. Let {Xi }∞ ANRs such that Xi is a i=1 be a sequence of locally compact nondegenerate ∏∞ X is a Q-manifold. compact AR for all but ﬁnitely many i. Prove that i i=1

§1.2. Homotopy dense and homotopy negligible sets. “Nice” completions of ANR’s Let X be a space. A set A ⊂ X is called homotopy dense if there exists a homotopy h : X × [0, 1] → X such that h0 = idX and h(X × (0, 1]) ⊂ A. A set A ⊂ X is called homotopy negligible provided its complement X\A is homotopy dense in X. An embedding e : Y → X is deﬁned to be homotopy dense (resp. homotopy negligible) if e(Y ) is a homotopy dense (resp. homotopy negligible) set in X. 1.2.1. Proposition. Suppose X is an A(N)R and Y ⊂ X is a homotopy dense set in X. Then Y is an A(N)R. Proof. Let h : X × [0, 1] → X be a homotopy such that h0 = id and h(X × (0, 1]) ⊂ Y . To prove that Y is an ANR we will apply Proposition

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1.1.1. Let A be a space, B a closed subset in A, and f : B → Y a map. Since X ⊃ Y is an ANR, the map f extends to a map f¯ : U → X of some neighborhood U of B in A. Let d be a metric on A, and let f˜ : U → X be a map deﬁned by f˜(a) = h(f¯(a), d(a, B)), a ∈ U . Obviously f˜(U ) ⊂ Y and f˜|B = f , i.e., f˜ : U → Y is the required extension of f . 1.2.2. Theorem. A subset A of X ∈ ANR is homotopy dense iﬀ for every cover U ∈ cov(X) and every map f : I n → X of a ﬁnite-dimensional cube with f (∂I n ) ⊂ A there exists a map f¯ : I n → A such that f¯|∂I n = f |∂I n and (f¯, f ) ≺ U. Proof. Fix any metric d on X and let d˜ be the metric on X × (0, 1] deﬁned ˜ by d((x, t), (x′ , t′ )) = max{d(x, x′ ), |t − t′ |}. Consider the open cover U0 = {B((x, t), t/3) | (x, t) ∈ X × (0, 1]} of X × (0, 1]. Our strategy is to construct a map f : X × (0, 1] → X × (0, 1] having the properties: f (X × (0, 1]) ⊂ A and (f, id) ≺ U0 . Once such a map f is constructed, then letting { h(x, t) =

x, if t = 0, prX ◦ f (x, t), if t ∈ (0, 1],

we deﬁne a homotopy h : X × [0, 1] → X with h(X × (0, 1]) ⊂ A (this of course means that A is homotopy dense in X). Pick a sequence of covers {Un }n≥1 ⊂ cov(X × (0, 1]) such that St Un+1 ≺ Un and mesh Un < 2−n for n ≥ 1. The ﬁrst step in constructing the map f is to ﬁnd a map f0 of the form p q0 f0 : X × (0, 1] → K → X × (0, 1], where K is a locally ﬁnite simplicial complex. The map f0 can be chosen so that (f0 , id) ≺ U1 , see 1.1.9. Denote by K (n) the n-skeleton of K. Using 1.1.11, ﬁnd a cover V ∈ cov(X × (0, 1]) such that any V-close to q0 |K (0) map g : K (0) → X × (0, 1] extends to a U2 -close to q0 map g¯ : K → X × (0, 1]. Using the hypothesis (in fact density of A in X) one can construct a map g : K (0) → A × (0, 1] such that (g, q0 |K (0) ) ≺ V. By the choice of V, the map g extends to a map q1 : K → X × (0, 1] such that (q1 , q0 ) ≺ U2 . Proceeding in this way, construct inductively a sequence of maps {qn : K → X × (0, 1]}n≥1 satisfying for every n ≥ 1 the conditions: qn+1 |K (n−1) = qn |K (n−1) , (qn+1 , qn ) ≺ Un+2 , qn+1 (K (n) ) ⊂ A × (0, 1]. Letting ﬁnally q = limn→∞ qn : K → X × (0, 1] and f = q ◦ p, we obtain the required map f : X × (0, 1] → X × (0, 1] with the properties f (X × (0, 1]) ⊂ A × (0, 1] and (f, id) ≺ U0 .

1.2. HOMOTOPY DENSE AND NEGLIGIBLE SETS

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1.2.3. Corollary. Suppose X ∈ AN R. If A is a homotopy dense (negligible) set in X then for every open set U ⊂ X the set U ∩ A is homotopy (negligible) dense in U . The following result is main in this section. 1.2.4. Theorem. Every ANR admits a homotopy dense embedding into a Polish ANR. Proof. Let X be an ANR. We can assume X to be a subset in the facet Q × {0} of a Hilbert cube Q × [0, 1]. Consider the union X ∪ Q × (0, 1]. Since X ∈ ANR is a closed subset in X ∪ Q × (0, 1], there is a retraction r : U → X of an open neighborhood U of X in X ∪ Q × (0, 1]. Letting V = Q × [0, 1]\ ClQ×[0,1] (Q × (0, 1]\U ) we enlarge U to a neighborhood V of X in Q × [0, 1]. ˜ → X ¯ of By 1.1.13, the retraction r : U → X extends to a map r˜ : U ˜ ¯ some Gδ -set U ⊂ V containing U (here X is the closure of X in Q × {0}). ˜ = U ˜ ∩X ¯ and remark that X, ˜ being a Gδ -set in X, ¯ is a Polish Let X ˜ is an ANR and (2) X is space as well. Now our aim is to show that (1) X ˜ homotopy dense in X. ˜ ∈ ANR remark that U ⊂ U ˜ ⊂ V and V \U ˜ ⊂ V \U ⊂ To show that X V ∩ (Q × {0}). The facet Q × {0} is a homotopy negligible set in Q × [0, 1]. ˜ is a homotopy negligible set in V . Hence by 1.2.3, V ∩ Q × {0} ⊃ V \U ˜ ˜ is an Therefore, U is a homotopy dense set in V ∈ ANR and by 1.2.1, U ANR. Now remark that r˜(x) = r(x) = x for all x ∈ X. Since X is dense in ˜ and r˜ is continuous, r˜(x) = x for all x ∈ X. ˜ Therefore, X ˜ being a retract X ˜ of U ∈ ANR, is an ANR. ˜ construct ﬁrstly a map To show that X is a homotopy dense set in X, ˜ → (0, 1] such that {(x, t) | 0 ≤ t ≤ ε(x)} ⊂ V (see Ex.4 to §1.1.A). By ε:X the choice of V , we have V ∩ Q × (0, 1] ⊂ U . Letting h(x, t) = r˜(x, t · ε(x)) ˜ × [0, 1] → X ˜ with h(X ˜ × (0, 1]) ⊂ X and we deﬁne a homotopy h : X ˜ h(x, 0) = x for all x ∈ X.

Exercises to §1.2. 1. Show that for every space X the set X × {0} is homotopy negligible in X × [0, 1]. More generally: show that for every homotopy dense set A ⊂ X and every Y the set A × Y is homotopy dense in X × Y . 2. Suppose A ⊂ A′ ⊂ X ′ ⊂ X and A is homotopy dense in X. Show that A′ is homotopy dense in X ′ . 3. Show that a subset A ⊂ X is homotopy dense in X if and only if each point x ∈ X has a neighborhood U such that U ∩ A is homotopy dense in U . 4. Suppose A, B are two homotopy negligible sets in X and A is closed. Show that the union A ∪ B is homotopy negligible in X. Show that the condition on A to be closed

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5. Show that a subset Y ⊂ X is homotopy dense if and only if given a space A, a closed subset B ⊂ A, a map f : A → X, and a cover U ∈ cov(X), there exists a map f¯ : A → X such that f¯|B = f |B, (f¯, f ) ≺ U , and f¯(A\B) ⊂ Y . 6. Let A be a homotopy dense set in X and d be a metric on X. Construct a homotopy h : X × [0, 1] → X with the properties: • h0 = id; • h(X × (0, 1]) ⊂ A; • d(h(x, t), x) ≤ t for every (x, t) ∈ X × [0, 1]. 7. Suppose X ⊂ Y ⊂ Z, X is homotopy dense in Y , and Y is homotopy dense in Z. Show that X is homotopy dense in Z. Hint: apply Ex.6 or Ex.5. 8. Suppose that Y is a homotopy dense set in X. Show that the inclusion Y ⊂ X is a ﬁne homotopy equivalence. A map f : Y → X is called a ﬁne homotopy equivalence if for every cover U ∈ cov(X) there is a map g : X → Y such that f ◦ g is U -homotopic to idX and g ◦ f is f −1 (U )-homotopic to idY . 10. Suppose Y ⊂ X are ANR’s. Show that the following conditions are equivalent: a) Y is homotopy dense in X; b) the embedding Y ⊂ X is a ﬁne homotopy equivalence; c) the set C(Q, Y ) is dense in C(Q, X). 11. Find a σ-compact set X ⊂ l2 such that C(Q, X) is dense in C(Q, l2 ) but X is not homotopy dense in l2 . We say that a subset A ⊂ X is LC ∞ rel. X if for every n ∈ N, every point x ∈ X and every neighborhood U ⊂ X of x there exists a neighborhood V ⊂ X of x such that every map f : ∂I n → V ∩ A admits an extension f¯ : I n → U ∩ X. A subset A ⊂ X is called strongly LC ∞ rel. X if for every point x ∈ X and every neighborhood U ⊂ X of x there exists a neighborhood V ⊂ X of x such that every map f : ∂I n → V ∩ A where n ∈ N admits an extension f¯ : I n → U ∩ X. A space is strongly LC ∞ , provided X is strongly LC ∞ rel. X. 12. Let X be an ANR. Show that for a subset A ⊂ X the following conditions are equivalent: a) A is homotopy dense in X; b) A is LC ∞ rel. X; c) A is strongly LC ∞ rel. X; d) X\A is locally homotopy negligible (i.e., for every n ≥ 0 and open U ⊂ X the relative homotopy group πn (U, U \A) vanishes); e) for every x ∈ X the space A ∪ {x} is LC ∞ ; f) every space Y with A ⊂ Y ⊂ X is an ANR. Hint: see H.Toru´ nczyk [1978]. 13. Suppose X is a subset of a linear metric space. Show that every convex dense subset in X is strongly LC ∞ rel. X. Let (X, d) be a metric space. A subset A ⊂ X is called uniformly strongly LC ∞ rel. X if for every ε > 0 there is δ > 0 such that for every point x ∈ X, and every n ∈ N any map f : ∂I n → B(x, δ) ∩ A admits an extension f¯ : I n → B(x, ε) ∩ A. A metric space (X, d) is called uniformly strongly LC ∞ if X is uniformly strongly LC ∞ rel. X. 14. Supposing that (X, d) is a uniformly strongly LC ∞ metric space, show that the com-

1.3. STRONG DISCRETE APPROXIMATION PROPERTY

17

ˆ of (X, d) is uniformly strongly LC ∞ and X is uniformly strongly LC ∞ ˆ d) pletion (X, ˆ rel. X. 15. Let K be a locally compact ∪AR and let K1 ⊂ · · · ⊂ Kn ⊂ . . . be a tower of compact subsets of K with K = Kn , Kn ⊂ Int Kn+1 , n ∈ N. Suppose K\Kn is n∈N contractible for every n and denote by K∞ the one-point compactiﬁcation of K. a) Show that K is homotopy dense in K∞ ; b) Show that K∞ is an AR; c) Show that K∞ ∼ = Q iﬀ K is a Q-manifold. Hint: see T.Dobrowolski [1982], H.Toru´ nczyk [1978]. 16. Let A be a homotopy dense subset of a space X. Show that A is an ANR if and only if X is an ANR. 17. Find a pair Y ⊂ X such that Y is an AR, X\Y is locally homotopy negligible in X, but X is not an AR. Hint: Consider the constructed by R.Cauty example of a σ-compact linear metric space X which is not an AR, and let Y ⊂ X be any dense linear subspace in X with countable Hamel basis. 18. For a subset A ⊂ X let W (X, A) = {(xn )n∈N ∈ X ω | xn ∈ A for almost all n} ⊂ X ω . If A = {∗} we let W (X, ∗) = W (X, {∗}). Suppose X is an AR, A a homotopy dense subset in X, and ∗ ∈ A. Show that the set W (A, ∗) is homotopy dense in X ω .

§1.3. The strong discrete approximation property In this section we discuss the strong discrete approximation property, the property characterizing s-manifolds among Polish ANR’s. The main result of this section (Theorem 1.3.2) characterizes ANR’s satisfying SDAP as exactly those spaces that admit a homotopy dense embedding into an s-manifold. A space X is deﬁned to satisfy the strong discrete approximation property (brieﬂy SDAP) if for every map f : Q × N → X and every cover U ∈ cov(X) there exists a map f¯ : Q×N → X such that (f¯, f ) ≺ U and {f¯(Q×{n})}n∈N is a discrete collection in X. The following characterization of SDAP is quite useful. 1.3.1. Proposition. A space X has SDAP if and only if every map f : Q × N → X can be approximated by maps sending {Q × {n}}n∈N onto a locally ﬁnite collection in X. Proof. The “only if” part is trivial. Suppose every map f : Q × N → X can be approximated by maps sending {Q × {n}}n∈N onto a locally ﬁnite collection in X. To prove that X has SDAP, ﬁx a map f : Q × N → X and a cover U ∈ cov(X). Denote by pr : Q × N × N → Q × N the projection onto the ﬁrst two factors and consider the map f ◦ pr : Q × N × N → X. According to our assumptions we can ﬁnd a map g : Q × N × N → X such

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that (g, f ◦ pr) ≺ U

(1) (2)

and

{g(Q × {(n, m)})}(n,m)∈N2 is a locally ﬁnite collection in X.

Remark that (1) implies (3)

for every (q, n, m) ∈ Q × N × N there is a U ∈ U with {g(q, n, m), f (q, n)} ⊂ U ,

while (2) implies that for every (n, m) ∈ N×N the set F(n, m) = {(n′ , m′ ) ∈ N × N | g(Q × {(n′ , m′ )}) ∩ g(Q × {(n, m)}) ̸= ∅} is ﬁnite. Let ξ(1) = 1 and inductively, using ﬁniteness of F(n, m)’s, for every n > 1 pick up a ξ(n) ∈ N to satisfy (n, ξ(n)) ∈ /

∪

F(k, ξ(k)).

k 0 the collection {f¯(Q × [t, 1] × {n})}n∈N is locally ﬁnite in X. Fix k ∈ N. The collection {f¯(Q × [ k1 , 1] × {n})}n∈N , being locally ﬁnite ˜ (see in X, is in fact, locally ﬁnite in some open neighborhood Uk of X in X Ex.5.c to §1.1.A). ∩ Write M = k∈N Uk . We claim that M is the required s-manifold containing X as a homotopy dense subset. Notice ﬁrstly that M is a Gδ -set in ˜ with X ⊂ M ⊂ X. ˜ Since X is homotopy dense in X, ˜ the Polish ANR X ˜ we get M is homotopy dense in X and X is homotopy dense in M (Ex.2 to §1.2). By Proposition 1.2.1, M is a Polish ANR. To show that M is an s-manifold, according to Characterization Theorem 1.1.14 and Proposition 1.3.1, it is enough to verify that every map g : Q × N → M can be approximated by a map sending {Q × {n}}n∈N onto a locally ﬁnite collection in M. Fix a cover U ∈ cov(M ) and a map g : Q × N → M which can be thought also as a sequence of maps {gk }k∈N ⊂ C(Q, M ). Since X is homotopy dense in M , C(Q, X) is dense in C(Q, M ); consequently the set {fn | n ∈ N} is dense in C(Q, M ). Replacing if necessary gk ’s by near maps, without loss of generality, we can assume that each gk = fn(k) for some n(k). Moreover, the numbers n(k) can be chosen pairwise distinct. Let ε : M → (0, 1] be a Lipschitz map such that for every x ∈ M there is U ∈ U with B(x, 2ε(x)) ⊂ U (see Ex.2 to §1.1.A). For every k ∈ N deﬁne a map g¯k : Q → M by g¯k (q) = f¯(q, ε ◦ gk (q), n(k)). It follows from the choice of the map ε that for every q ∈ Q we have d(¯ gk (q), gk (q)) = d(f¯(q, ε ◦ gk (q), n(k)), f (q, n(k))) ≤ ε ◦ gk (q) < 2ε ◦ gk (q), and consequently, (¯ gk , gk ) ≺ U. Let us show that the collection {¯ gk (Q)}k∈N is locally ﬁnite in M . Fix any point x0 ∈ M and consider its neighborhood U = {x ∈ M | ε(x) > ε(x0 )/2}. Remark that if for some q ∈ Q and k ∈ N we have g¯k (q) ∈ U then ε ◦ gk (q) ≥ ε(x0 )/4 (this easily follows from the inequalities: |ε ◦ g¯k (q) − ε ◦ gk (q)| ≤ d(¯ gk (q), gk (q)) ≤ ε ◦ gk (q) and ε ◦ g¯k (q) > ε(x0 )/2). Hence for every k ∈ N we have g¯k (Q) ∩ U ⊂ f¯(Q × [ε(x0 )/4, 1] × {n(k)}). Since the

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collection {f¯(Q × [e(x0 )/4, 1] × {n})}n∈N is locally ﬁnite in M , the point x0 has a neighborhood V ⊂ U intersecting only ﬁnitely many of the sets f¯(Q × [ε(x0 )/4, 1] × {n}). Since the numbers n(k) are pairwise distinct, V meets only a ﬁnitely many of the sets g¯k (Q), i.e., the collection {¯ gk (Q)}k∈N is locally ﬁnite in M . Proof of Claim. Since X is an ANR, there is a cover U ∈ cov(X) such that any two U-close maps into X are 14 -homotopic. Let f0 = f ◦ pr : Q × I × N → X, where pr : Q × I × N → Q × N is the projection. Since X has SDAP, there exists a map f ′ : Q × I × N → X such that (f0 , f ′ ) ≺ U and the family {f ′ (Q × I × {k})}k∈N is locally ﬁnite in X. Since the maps f0 and f ′ are 41 -homotopic, there is a map f1 : Q × I × N → X such that f1 = f0 on Q × [0, 12 ] × N, f1 = f ′ on Q × {1} × N and d(f1 (q, t, k), f (q, k)) = d(f1 (q, t, k), f0 (q, t, k)) ≤ 14 for every (q, t, k) ∈ Q × I × N. By induction, we will construct a sequence of maps {fn : Q × I × N → X}n∈N satisfying the following conditions: 1 1 (1n+1 ) fn+1 = fn on Q × ([0, 2n+1 ] ∪ [ 2n−1 , 1]) × N; 1 (2n+1 ) d(fn+1 (q), fn (q)) < 2n+2 for any q ∈ A × I × N; (3n+1 ) the family {fn+1 (Q × [ 21n , 1] × {k})}k∈N is locally ﬁnite in X. Inductive construction. Suppose the map fn : Q × I × N → X satisfying the conditions (1n ) − (3n ) has been constructed. Since the family 1 {fn (Q×[ 2n−1 , 1]×{k})}k∈N is locally ﬁnite in X, by Ex.2 to §1.1.A, there is a 1 cover U ∈ cov(X) such that mesh U < 2n+2 and the collection {St (fn (Q × 1 [ 2n−1 , 1] × {k}), St U)}k∈N is locally ﬁnite in X. For every k ∈ N ﬁx a 1 neighborhood Wk ⊂ Q × I × {k} of Q × [ 2n−1 , 1] × {k} such that fn (Wk ) ⊂ ∪ 1 St (fn (Q × [ 2n−1 , 1] × {k}), U). Set W = k∈N Wk . Since X is an ANR, there is a cover V ∈ cov(X) such that any two V-close maps into X are Uhomotopic. SDAP of X supplies us with a map f ′ : Q×I ×N → X such that (f ′ , fn ) ≺ V and the family {f ′ (Q×I ×{k})}k∈N is locally ﬁnite in X. Since the maps fn and f ′ are U-homotopic, there is a map fn+1 : Q × I × N → X 1 such that (fn , fn+1 ) ≺ U and fn+1 (q) = fn (q) if q ∈ Q × ([0, 2n+1 ]∪ 1 1 ′ [ 2n−1 , 1]) × N, and fn+1 (q) = f (q) if q ∈ (Q × [ 2n , 1] × N)\W . It is easily seen that the map fn+1 satisﬁes the conditions (1n+1 ) and (2n+1 ). To see that the family fn+1 (Q × [ 21n , 1] × {k})}k∈N is locally ﬁnite in X, notice that for every k ∈ N we have fn+1 (Q × [ 21n , 1] × {k}) ⊂ f ′ (Q × I × {k}) ∪ 1 St (fn (Q × [ 2n−1 , 1] × {k}), St U). Since the collections {f ′ (Q × I × {k})}k∈N 1 and {St (fn (Q × [ 2n−1 , 1] × {k}), St U)}k∈N are locally ﬁnite in X, so is the collection {fn+1 (Q × [ 21n , 1] × {k})}k∈N . The inductive step is over. One can readily verify that the map f¯ = limn→∞ fn : Q × I × N → X satisﬁes the conditions (1) and (2).

1.3. STRONG DISCRETE APPROXIMATION PROPERTY

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1.3.3. Corollary. If X is an ANR with SDAP then every open subspace in X satisﬁes SDAP too. Proof. Using Theorem 1.3.2, ﬁnd an s-manifold M containing X as a ho˜ motopy dense subset. Given an open subspace U ⊂ X ﬁnd an open set U ˜ ∩ X = U . By 1.2.3, U = U ˜ ∩ X is a homotopy dense in M such that U ˜ . The space U ˜ , being open in M , is an s-manifold. Applying subset in U 1.3.2 again, we get U is an ANR with SDAP. 1.3.4. Theorem. For an X ∈ ANR the following conditions are equivalent: a) X has SDAP; b) every map f : K → X of a locally compact space can be approximated by a perfect map; c) for every space F and a locally ﬁnite collection {Fi }i∈I of subsets in F every map f : F → X can be approximated by a map sending {Fi }i∈I onto a locally ﬁnite collection in X. Proof. To prove the implication c)⇒b) ∪ ﬁx a map f : K → X of a locally compact space K and write K = n∈N Kn , where Kn ⊂ Int Kn+1 are compact subsets of K. Notice that the collection {K\Kn }n∈N is locally ﬁnite in K. Hence by c), the map f can be approximated by a map f¯ sending {K\Kn }n∈N onto a locally ﬁnite collection in X. One can easily show that then the map f¯ is perfect. The implication b)⇒a) follows from Proposition 1.3.1. To prove the implication a)⇒c) ﬁx a cover U ∈ cov(X) and a map f : ˜ containing X F → X. By Theorem 1.3.2, there exists an s-manifold M as a homotopy dense set. Using Ex.1 to §1.1, ﬁnd an open neighborhood ˜ of X and a cover U˜ ∈ cov(M ) such that U˜ ∩ X = U. Notice that M M ⊂M is an s-manifold. By Lavrentiev Theorem 1.1.13, there are a Polish space F ′ containing F and a map f ′ : F ′ → M extending the map f . Since the collection {Fi }i∈I is locally ﬁnite in F , there is an open neighborhood F˜ of F in F ′ such that the family {Fi }i∈I is locally ﬁnite in F˜ . Let ﬁnally ˜ f˜ = f ′ |F˜ : F˜ → M . Let V ∈ cov(M ) be a cover such that St V ≺ U. ˜ According to 1.1.21, there is a closed embedding e : F → M such that (f˜, e) ≺ V. Then the collection {e(Fi )}i∈I is locally ﬁnite in M . By Ex.3 to §1.1.A, there is a cover W ∈ cov(M ) such that W ≺ V and the family {St (e(Fi ), W)}i∈I is locally ﬁnite in M . Using homotopy density of X in M , ﬁnd a map f¯ : F → X with (f¯, e|F ) ≺ W. It is easy to verify that the map f¯ is as required, i.e. (f¯, f ) ≺ U and the collection {f¯(Fi )}i∈I is locally ﬁnite in X. Exercises and Problems to §1.3. 1. Verify that both l2 and s satisfy SDAP.

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2. Suppose X is a non-compact AR. Show that the countable power X ω satisﬁes SDAP. 3. Show that for a space X the following conditions are equivalent: a) X has SDAP; b) every map f : ⊕n∈N I n → X can be approximated by maps sending {I n }n∈N onto a discrete family in X; c) every map f : ⊕n∈N I n → X can be approximated by maps sending {I n }n∈N onto a locally ﬁnite family in X; 4. Suppose X is an ANR with SDAP. Show that every homotopy dense subspace in X satisﬁes SDAP. 5. Let X be an ANR with SDAP. Show that every ANR is homotopically equivalent to a certain open subspace of X. 6. Show that every ANR with SDAP is inﬁnite-dimensional. 7. Use Triangulation Theorem 1.1.16 and ANR-Theorem 1.1.24 to show that every ANR X satisfying SDAP admits an embedding X ⊂ M into a Q-manifold M such that X is both homotopy dense and homotopy negligible in M . 8. Suppose X is an ANR with SDAP. Show that for every ANR Y the product X × Y satisﬁes SDAP. 9. Suppose X is a space and K is a locally compact space such that X × K has SDAP. Show that X satisﬁes SDAP. 10. Prove 1.3.3 and 1.3.4 (without applying the theory of s-manifolds). 11. A space X is deﬁned to satisfy the discrete n-cells property if every map f : I n × N → X approximates by maps sending {I n × {i}}i∈N onto a discrete family in X. Find an example of a Polish AR X which satisﬁes the discrete n-cells property for every n ∈ N, but does not satisfy SDAP. Hint: For every n identify the n-dimensional cube I n with the subset [2n, 2n + 1] × I n−1 ⊂ Rn∪of l2 and consider the one-point∪ compactiﬁcation αA = {a0 } ∪ A of the space ∞ ∞ A = R1 ∪ n=1 I n ⊂ l2 . Let ∂A = R1 ∪ n=1 ∂I n and K = αA × {0} ∪ ∂A × [0, 1]. The required space X can be obtained as a suitable metrization of the quotient space K × s/{(a0 , 0)} × s. For details, see M.Bestvina et al [1986]. 12. A space X is deﬁned to satisfy the locally compact approximation property (brieﬂy LCAP) if for every cover U ∈ cov(X) there exists a map f : X → X such that (f, idX ) ≺ U and ClX (f (X)) is locally compact. a) Show that every locally compact space satisﬁes LCAP. b) Show that an ANR X satisﬁes LCAP if and only if for every cover U ∈ cov(X) there are a locally ﬁnite simplicial complex K, a map p : X → K, and a perfect map q : K → X such that (q ◦ p, idX ) ≺ U . c) Show that every open subspace of an ANR with LCAP satisﬁes LCAP too. d) Show that every homotopy dense subspace of an ANR with LCAP satisﬁes LCAP too. e) Show that every ANR with SDAP satisﬁes LCAP. f) Suppose X is an ANR with LCAP. Show that X satisﬁes SDAP if and only if X satisﬁes the discrete n-cells property for every n ∈ N. g) Suppose X is an ANR with LCAP and Y ⊂ X a homotopy negligible subset such that C(Q, Y ) is dense in C(Q, X). Show that Y satisﬁes SDAP. h) Suppose X is an ANR with LCAP and Y ⊂ X a subset which is both homotopy dense and homotopy negligible in X. Show that Y satisﬁes SDAP.

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i) Suppose X satisﬁes LCAP. Show that for every compactum K the product K × X satisﬁes LCAP too. j) Show that X satisﬁes LCAP if and only if there is a homotopy h : X × [0, 1] → X such that for every t ∈ [0, 1] we get d(h(x, t), x) ≤ t and h(X × [t, 1]) has locally compact closure in X. Hint: see T.Banakh [199?b] 13. (Open Problem) Characterize spaces admitting homotopy dense embeddings into locally compact ANR’s. In particular, does every ANR with LCAP admit a homotopy dense embedding into a locally compact ANR? 14. Find an example of an AR which admits no homotopy dense ∪ embedding into a locally compact ANR. Hint: Consider the “hedgehog” space H = n∈N [0, en ] ⊂ l2 , where (en ) stands for the standard orthonormal basis of l2 .

§1.4. Z-Sets and Strong Z-Sets In this section we study in detail Z-sets, which were introduced in the ﬁrst section. We recall the deﬁnition. A subset A of a space X is called a (strong) Z-set if it is closed and for every cover U ∈ cov(X) there is a map f : X → X such that (f, idX ) ≺ U and f (X) ∩ A = ∅ (ClX (f (X)) ∩ A = ∅). 1.4.1. Proposition. Every Z-set in a locally compact space is strong. Proof. Let X be a locally compact space and A a Z-set in X. To show that A is a strong Z-set in X, ﬁx a cover U ∈ cov(X). Since the identity map idX : X → X is perfect, by Ex.7 to §1.1.A, there is a cover W ∈ cov(X), W ≺ U , such that every map p : X → X, W-near to idX , is perfect. Using the fact that A is a Z-set in X, ﬁnd a map p : X → X with (p, idX ) ≺ W ≺ U and p(X) ∩ A = ∅. By the choice of W, the map p is perfect, and consequently p(X) is closed in X. Then p : X → X is a map with (p, idX ) ≺ U and ClX (p(X)) ∩ A = ∅, i.e., A is a strong Z-set in X. The locally compact condition in the above proposition is essential because of the following simple ∪ 1.4.2. Example. Consider the space X = [0, 1] × {0} ∪ n∈N { n1 } × [0, 1] ⊂ R2 and observe that X is a Polish space which is locally compact at each point excepting (0, 0). This space is one-dimensional, contractible and locally contractible, and hence is an AR. Furthermore, the one-point set A = {(0, 0)} is a Z-set in X but not a strong Z-set. In the meantime, we have the following useful 1.4.3. Proposition. Suppose X is an ANR with SDAP. Then every Z-set in X is strong.

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Proof. Let A be a Z-set in X. Since X is an ANR, we may approximate q p the identity map X → X by a map X → K → X, where K is a locally ﬁnite simplicial complex, see 1.1.9. Using Theorem 1.3.4, replace p by a close perfect map p′ : K → X. Then by Ex.7 to §1.1.A, any suﬃciently close to p′ map p′′ : K → X is perfect. If f : X → X is a suﬃciently close to idX map with f (X) ∩ A = ∅ then the map f ◦ p′ is close to p′ , and thus is perfect. This yields f ◦ p′ (K) is a closed set in X with f ◦ p′ (K) ∩ A = ∅. Then the map f¯ = f ◦ p′ ◦ q : X → X is close to idX and has the property ClX (f¯(X)) ⊂ f ◦ p′ (K). This gives ClX (f¯(X)) ∩ A = ∅, i.e., A is a strong Z-set in X. A closed subset A of a space X is deﬁned to be Z∞ -set if for every U ∈ cov(X) and every map f : I n → X of a ﬁnite-dimensional cube there is a map f¯ : I n → X such that (f¯, f ) ≺ U and f¯(I n ) ∩ A = ∅. 1.4.4. Theorem. Suppose X is an ANR. For a closed subset A ⊂ X the following conditions are equivalent. a) A is a Z-set in X; b) A is a Z∞ -set in X; c) A is homotopy negligible in X. Proof. The implications c) ⇒ a) ⇒ b) are trivial. Suppose A is a Z∞ -set in X. To prove that A is homotopy negligible in X we will apply Theorem 1.2.2. Fix U ∈ cov(X) and a map f : I n → X of a ﬁnite-dimensional cube with f (∂I n ) ⊂ X\A. Let U be a neighborhood of ∂I n in I n such that f (U ) ∩ A = ∅. We may assume that U is so ﬁne that St (f (U ), U) ∩ A = ∅. By 1.1.11, there is a cover V ∈ cov(X) such that every V-close to f map g : ∂I n ∪ I n \U → X has an extension g¯ : I n → X U-close to f . Since A is a Z∞ -set in X, there is a map g : I n → X\A such that (g, f ) ≺ V. By the choice of V there is a map f¯ : I n → X such that f¯|∂I n = f , f¯|I n \U = g, ¯ ), U) ∪ g(I n ), we get f¯(I n ) ∩ A = ∅. and (f¯, f ) ≺ U. Since f¯(I n ) ⊂ St (f (U Applying 1.2.2, we conclude A is homotopy negligible in X. Theorem 1.4.4 implies 1.4.5. Corollary. If A is a Z-set in an ANR X then for every open set U ⊂ X the intersection A ∩ U is a Z-set in U . Now we present a characterization of strong Z-sets in ANR’s. 1.4.6. Theorem. Let X be an ANR and d be a metric on X. A closed subset A of X is a strong Z-set in X if and only if there is a homotopy h : X × [0, 1] → X such that i) d(h(x, t), x) ≤ t for (x, t) ∈ X × [0, 1]; ii) for every t ∈ (0, 1] ClX (h(X × [t, 1])) ∩ A = ∅.

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Proof. Suppose A is a strong Z-set in X. The required homotopy h : X×I → X will be constructed inductively similarly as in Theorem 1.3.2. Let h0 = prX : X×I → X be the projection. Since X is an ANR, there is a cover U ∈ cov(X) such that any two U-near maps into X are 41 -homotopic. Since A is a strong Z-set in X, there is a map h′ : X×I → X such that (h0 , h′ ) ≺ U and ClX (h′ (X × I)) ∩ A = ∅. By the choice of U, the maps h0 and h′ are 1 4 -homotopic. Using this fact, one can construct a map h1 : X × I → X such that h1 |X × [0, 21 ] = h0 |X × [0, 12 ], h1 |X × {1} = h′ |X × {1}, and d(h1 (x, t), x) = d(h1 (x, t), h0 (x, t)) ≤ 41 for every (x, t) ∈ X × I. By induction, we will construct a sequence of maps {hn : X ×I → X}n≥1 satisfying the following conditions: 1 1 (1n+1 ) hn+1 (x, t) = hn (x, t) for all x ∈ X and t ∈ [0, 2n+1 ] ∪ [ 2n−1 , 1], 1 (2n+1 ) d(hn+1 (x, t), hn (x, t)) < 2n+2 for (x, t) ∈ X × I, (3n+1 ) ClX (hn+1 (X × [ 21n , 1])) ∩ A = ∅.

Inductive construction. Assume that the map hn : X × I → X satisfying (1n ) − (3n ) has been constructed. Using (3n ), ﬁnd two open sets U, W such 1 ¯ ⊂W ⊂W ¯ ⊂ X\A. Pick up a cover U ∈ that hn (X × [ 2n−1 , 1]) ⊂ U ⊂ U −n−2 ¯ }. By 1.1.10, there is a cov(X) such that mesh U < 2 and U ≺ {W, X\U cover V ∈ cov(X) such that any two V-near maps into X are U-homotopic. Since A is a strong Z-set in X, there is a map h′ : X × I → X such that (h′ , hn ) ≺ V and h′ (X × I) ∩ A = ∅. By the choice of V, the maps h′ and hn are U-homotopic. Using this fact, construct a map hn+1 : X × I → X such that (hn+1 , hn ) ≺ U and { hn+1 (x, t) =

1 1 hn (x, t), if x ∈ X and t ∈ I\( 2n+1 , 2n−1 ),

h′ (x, t),

if (x, t) ∈ X × ([2−n , 1])\hn (U ).

It is easily seen that the map hn+1 satisﬁes the conditions (1n+1 ) and (2n+1 ). To verify (3n+1 ) notice that hn+1 (X × [ 21n , 1]) ⊂ h′ (X × I) ∪ W and (h′ (X × I) ∪ W ) ∩ A = ∅. The inductive step is over. Letting h = limn→∞ hn : X × I → X we get the required homotopy satisfying the conditions (i)–(ii). Now suppose that for a closed subset A ⊂ X there is a homotopy h : X × I → X satisfying the conditions (i)–(ii). To show that A is a strong Z-set, ﬁx a cover U ∈ cov(X). By Ex.2 to §1.1.A, there is a Lipschitz map ε : X → (0, 1] such that {B(x, 2ε(x))}x∈X ≺ U. Consider the map f : X → X deﬁned by f (x) = h(x, ε(x)). By (i), d(f (x), x) = d(h(x, ε(x)), x) ≤ ε(x), and consequently, (f, id) ≺ U . Let us show that f (X) ∩ A = ∅. Assuming the converse, ﬁnd a point a ∈ f (X) ∩ A and consider its neighborhood U = {x ∈ X | d(x, a) < ε(a)/2}. Since the function ε is Lipschitz, ε(x) > ε(a)/2

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for every x ∈ U . Then f (X) ∩ U ⊂ h(X × [ ε(a) 2 , 1]), and consequently, a ∈ f (X) ∩ U ⊂ h(X × [ ε(a) 2 , 1]). But this contradicts to (ii). We deﬁne a map f : X → Y to be closed over a subset A ⊂ Y if for each a ∈ A and each neighborhood U of f −1 (a) (which might be empty) there exists a neighborhood V of a such that f −1 (V ) ⊂ U . We derive from Theorem 1.4.6 the following important statement that will be used very often throughout this monograph. 1.4.7. Corollary. Let A be a strong Z-set in an ANR X, f : C → X a map, and B a closed subset of C such that f |B : B → X is a closed embedding. Then for every U ∈ cov(X) there exists a map g : C → X such that (f, g) ≺ U, g|B = f |B, g(C\B) ∩ A = ∅, and g is closed over A. Proof. Let d be a metric on X and h : X × I → X the homotopy satisfying the conditions (i)–(ii) of Theorem 1.4.6. Pick a map ε : X → (0, 1] such that {B(x, 2ε(x))}x∈X ≺ U. Fix any metric ρ on C and deﬁne a map δ : C → [0, 1] by the formula δ(c) = min{ε ◦ f (c), ρ(c, B)}. The required map g : C → X can be deﬁned by the formula g(c) = h(f (c), δ(c)). 1.4.8. Proposition. Let A be a closed subset of X ∈ ANR. Then A is a Z-set in X if and only if there is a U ∈ cov(X) such that A ∩ U is a Z-set in U for each U ∈ U. Proof. Let U ∈ cov(X) be such that U ∩A is a Z-set in U for each U ∈ U. Let f : Q → X be any map and ε > 0, choose a ﬁnite subcover {U1 , . . . , Un } ⊂ U of the compactum f (Q). For every i ≤ n ﬁx a cover Ui ∈ cov(Ui ) such that mesh Ui < ε/n and Un ≺ {B(x, d(x, X\Ui ))}x∈Ui . Since A ∩ Ui is a Z-set in Ui , there is a map gi : Ui → Ui such that (gi , idUi ) ≺ Ui and gi (Ui )∩A = ∅. By the choice of Ui , the map gi extends to a map g¯i : X → X with g¯i |X\Ui = id. Letting f¯ = g¯n ◦ g¯n−1 ◦ · · · ◦ g¯1 ◦ f we get the map f¯ : Q → X such that d(f¯, f ) < ε and f (Q) ∩ A = ∅, i.e., A is a Z-set in X. 1.4.9. Proposition. If X is an ANR with SDAP then each compactum in X is a strong Z-set. Proof. Fix a compactum K ⊂ X. According to Theorem 1.4.3, it is enough to show that K is a Z-set in X. Fix a map f : Q → X and ε > 0. Consider the map f ◦ pr : Q × N → X, where pr : Q × N → Q is the natural projection. SDAP of X supplies us with a map g : Q × N → X such that d(g, f ◦ pr) < ε and the collection {g(Q × {n})}n∈N is discrete in X. Consequently, only a ﬁnite many of g(Q × {n})’s meets A. Thus there is an n0 with g(Q × {n0 }) ∩ A = ∅. Letting f¯(q) = g(q, n0 ) we get a map f¯ : Q → X such that d(f¯, f ) < ε and f¯(Q) ∩ A = ∅.

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27

A subset A ⊂ X ∪ is deﬁned to be a (strong) Zσ -set in X, provided A can be written as A = n∈N An , where each An in a (strong) Z-set in X. A space X is called a (strong) Zσ -space, provided X is a (strong) Zσ -set in X. 1.4.10. Theorem. Each strong Zσ -space X ∈ AN R satisﬁes SDAP. ∪ Proof. Write X = n∈N Xn , where each Xn is a strong Z-set in X. It can be shown that each compact subset of X is a strong Z-set in X. Let f : Q × N → X be a map and let U0 ∈ cov(X). We construct sequences {gn : X → X}, {Un | Un ∈ cov(X)}, and {(Vn , Wn ) | Vn , Wn are open sets in X, Vn ⊃ Wn ⊃ Wn ⊃ Xn ∪ gn ◦ gn−1 ◦ · · · ◦ g1 ◦ f (Q × {1, . . . , n})} such that gn (X) ∩ Vn = ∅, St Un ≺ Un−1 , St Un ≺ {Vn−1 , X\Wn−1 }, and (gn , idX ) ≺ Un . Deﬁne f¯ : Q × N → X by f¯(q, n) = gn ◦ gn−1 ◦ · · · ◦ g1 ◦ f (q, n) for (q, n) ∈ Q × N. Then f¯ is U0 -close to f , and {f¯(Q × {n})}n∈N is a discrete family in X (since f¯(Q × {1, . . . , n}) ∩ Wn = ∅, n ∈ N).

Exercises to §1.4. 1. Suppose {Ai }i∈I is a locally ﬁnite family of Z-sets in X ∈ ANR. Show that A = ∪ A is a Z-set in X. i∈I i 2. Suppose a closed topologically complete subset A of X ∈ ANR is the union A = ∪ A of Z-sets. Show that A is a Z-set in X. n n∈N 3. Show that every Zσ -set in a Polish ANR is homotopy negligible. 4. Suppose A is a closed subset of X ∈ ANR, B ⊂ A is a Z-set in X, and A\B is a Z-set in X\B. Show that A is a Z-set in X. 5. Let f : C → X be a map and B a closed subset in C such that f |B : B → X is a closed embedding and f (C\B) ∩ f (B) = ∅. Show that f is a closed embedding if and only if f : C\B → X\f (B) is a closed embedding and f is closed over f (B). 6. Let f : C → X be an injective map and {Ci }i∈I a collection of closed subsets in C such that the family {f (Cn )}i∈I is locally ﬁnite in X and for every i ∈ I the restriction f |Cn : Cn → X is a Z-embedding. Prove that f is a Z-embedding. 7. Let X, Y be ANR’s. Show that a subset A ⊂ X is a Z-set in X if and only if A × Y is a Z-set in X × Y . 8. Suppose∪ f : C → X is an injective map into an ANR X. Suppose C can be expressed as C = n∈N Cn , where each Cn is closed in C, and the collection {f (C\Cn )}n∈N is locally ﬁnite in X. Prove that f is a closed embedding (resp. a Z-embedding), provided for every n ∈ N the map f |Cn : Cn → X is a closed embedding (resp. a Z-embedding). 9. Suppose A is a strong Z-set in X ∈ ANR. Show that for every open set U in X U ∩ A is a strong Z-set in X. 10. Show that for a closed subset A of an ANR X the following conditions are equivalent: a) A is a strong Z-set in X;

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BASIC THEORY b) there exists a closed over A homotopy h : X × I → X such that h0 = idX and h(X × (0, 1]) ∩ A = ∅; c) for every map f : Q × N → X and every cover U ∈ cov(X) there is a map f¯ : Q × N → X such that (f¯, f ) ≺ U and ClX (f¯(Q × N)) ∩ A = ∅;

d) for every map f : ⊕n∈N I n → X and every cover U ∈ cov(X) there is a map f¯ : ⊕n∈N I n → X with (f¯, f ) ≺ U and ClX (f¯(⊕n∈N I n )) ∩ A = ∅.

11. Show that a subset A of X ∈ ANR is a strong Z-set in X if and only if A is both a Z-set and a strong Zσ -set in X. 12. Suppose ∪ {Ai }i∈I is a locally ﬁnite collection of strong Z-sets in X ∈ ANR. Show that A = i∈I Ai is a strong Z-set in X. 13. Show that a closed subset A of X ∈ ANR is a strong Z-set in X if and only if there is a U ∈ cov(X) such that A ∩ U is a strong Z-set in U for each U ∈ U . 14. Suppose A is a strong Z-set in an ANR X. Show that for any ANR Y the product A × Y is a strong Z-set in X × Y . 15. Find two ANR’s X, Y and a subset A ⊂ X such that A × Y is a strong Z-set in X × Y , but A is not a strong Z-set in X. 16. Suppose X is a homotopy dense subset in Y ∈ ANR and A is a closed subset in Y . Show that A is a Z-set in Y if and only if A ∩ X is a Z-set in X. 17. Suppose X is a homotopy dense subset in Y ∈ ANR and A is a strong Z-set in Y . Show that A ∩ X is a strong Z-set in X. ˜ ∈ ANR of X such 18. Let A be a strong Z-set in X ∈ ANR. Construct a completion X ˜ and Cl ˜ (A) is a strong Z-set in X. ˜ that X is homotopy dense in X, X

19. Give an example of an ANR Y , a homotopy dense subset X ⊂ Y , and a strong Z-set A in X such that ClY (A) is not a strong Z-set in Y . 20. Show that X ∈ ANR is a strong Zσ -space if and only if X is a Zσ -space satisfying SDAP. 21. Find an X ∈ AR that is a Zσ -space but not a strong Zσ -space. Hint: Use Ex.11 to §1.3. 22. Suppose X is an ANR with LCAP. Show that every Z-set in X is strong. 23. Find an X ∈ AR such that every Z-set in X is strong but X does not satisfy LCAP. Hint: See Ex.14 to §1.3. 24. A space X is deﬁned to be a co-Zσ -space, provided X contains a homotopy dense topologically complete subset G ⊂ X. Show that for an ANR X the following conditions are equivalent: a) X is a co-Zσ -space; b) for every Y that contains X as a homotopy dense subset the complement Y \X is contained in a Zσ -subset of Y ; c) there is a Polish ANR-space Y ⊃ X such that Y \X ⊂ A for certain Zσ -set A in Y. 25. Show that every co-Zσ -space X ∈ ANR satisfying SDAP contains a closed subset homeomorphic to s, and consequently, is strongly inﬁnite-dimensional (see §2 of Chapter II for the deﬁnition). 26. Let X be a co-Zσ -space. Show that every open subspace of X is co-Zσ . 27. Show that a space X is a co-Zσ -space if and only if each point x ∈ X has a neighborhood which is a co-Zσ -space. 28. Show that every co-Zσ -space is Baire.

1.5. STRONG UNIVERSALITY

29

29. Let X be an AR, A ⊂ X, and ∗ ∈ A. ¯ ̸= X. Show that W (X, A) is strong Zσ -space; a) Suppose A b) Suppose A is a Zσ -set in X. Show that W (X, A) is a strong Zσ -space; c) Suppose A is homotopy dense in X and A is a co-Zσ -space. Show that W (X, A) is a co-Zσ -space. 30. Let X be an AR such that X ω is a (co)Zσ -space. Is then X a (co)Zσ -space? 31. Suppose that X is an ANR such that every point of X is a Z-set in X. Prove that a countably dimensional subset A ⊂ X is a Z-set in X if and only if A is a Z2 -set in X (the latter means that each map [0, 1]2 → X can be uniformly approximated by a map [0, 1]2 → X \ A). 32. Let X, Y be LC ∞ -spaces, A ⊂ X be a Zn -set in X and B ⊂ Y be a Zm -set in Y . Prove that A × B is a Zn+m+1 -set in X × Y . 33. Supposet that X is an LC ∞ -space of the ﬁrst Baire category and X is homeomorphic to X × X. Prove that X is a σZn -space for every n ∈ N.

§1.5. Strong Universality. In this section we introduce the conception of a strongly universal space, the conception playing a crucial role in this book. • We deﬁne a space X to be strongly C-universal, where C is a space, if for every cover U ∈ cov(X), every closed subset B ⊂ C, and every map f : C → X that restricts to a Z-embedding on B, there exists a Z-embedding f¯ : C → X such that f¯|B = f |B and (f¯, f ) ≺ U. • A space X is called strongly C-universal, where C is a class of spaces if X is strongly C-universal for every C ∈ C. • A space X is called strongly universal if it is strongly X-universal. First we will prove that the strongly C-universal property of ANR’s is local. We say that a property P of topological spaces is local if a space X satisﬁes P if and only if each point x ∈ X has a neighborhood satisfying P. A class of spaces C is called closed-hereditary, provided every closed subspace of each C ∈ C belongs to C. 1.5.1. Proposition. An ANR X is strongly C-universal, where C is a closed-hereditary class of spaces, if and only if each point x in X has a strongly C-universal neighborhood. For the proof we apply the following fundamental theorem on local properties due to E.Michael [1954] (see also C.Bessaga, A.Pelczy´ nski [1975]). 1.5.2. Theorem. A property P of spaces is local if P satisﬁes the following conditions: a) if X has P then every open subspace U of X has P; b) if X is the union of two open sets both of which have the property P then X has P;

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c) if X is the union of a disjoint collection of open sets all of which have the property P, then X has the property P. Therefore, to prove Proposition 1.5.1, it is enough to prove the following three lemmas. 1.5.3. Lemma. Suppose X is a strongly C-universal ANR for a closedhereditary class C. Then every open subspace U ⊂ X is strongly C-universal. Proof. Let d be a metric on X, U an open subset of X, and f : C → U a map of a space C ∈ C into U such that ∪ f |D : D → U is a Z-embedding for some closed D ⊂ C. We have U = n∈N Un , where Un = {x ∈ X | −n −1 −1 d(x, X\U ∪ ) ≥ 2 }. Let An = f (Un ) and Bn = f (U \ Int Un+1 ). Then C = n∈N An , An ⊂ Int An+1 , and An , Bn are disjoint closed subsets of C. For a given map ε : U → (0, 1) we shall construct a sequence {fn : C → U } satisfying the following conditions: (i) fn |Bn ∪ D = f |Bn ∪ D, (ii) fn |An ∪ D : An ∪ D → U is a Z-embedding, (iii) fn |An−1 ∪ Bn ∪ D = fn−1 |An−1 ∪ Bn ∪ D, (iv) fn is εn -close to fn−1 , where εn : X → (0, 1) is a map such that εn (x) = 2−n min{ε(x), d(x, X\U )} for x ∈ Un+1 , and (v) fn (An+2 ) ⊂ Un+2 . Without loss of generality, A0 = ∅ and B0 = C, so we can set f0 = f . Let us assume that fn−1 has been constructed. Since X is a strongly C-universal ANR, there is an εn -homotopy gt : C → X such that g0 = fn−1 , g1 : C → X is a Z-embedding, and g1 |An−1 ∪ (D\ Int Bn ) = fn−1 |An−1 ∪ (D\ Int Bn ). We can also assume that gt (c) = f (c) for each c ∈ D\ Int Bn . Deﬁne fn : C → X by fn (c) = gλ(c) (c), where λ : C → [0, 1] is a Urysohn function satisfying λ(Bn ) = {0}, λ(An ) = {1}. To check (v) let c ∈ An+2 . If c ∈ An+1 , then fn−1 (c) ∈ Un+1 and hence, by (iv), 1 d(fn−1 (c), X\U ) ≤ d(fn (c), X\U ), 2 which implies d(fn (c), X\U ) ≥ 12 2−(n+1) and thus fn (c) ∈ Un+2 . If c ∈ An+2 \An+1 ⊂ Bn , then fn (c) = f (c) ∈ Un+2 . Consequently, (v) holds and it implies fn (C) ⊂ U . Deﬁne a map h : C → U by h = limn→∞ fn . It is clear that h is ε-close to f . Also, (ii), (iii) and (iv) imply that h is an embedding. Note that h(C) =

∞ ∪ n=0

h(An+1 \ Int An ) =

∞ ∪

fn+1 (An+1 \ Int An ),

n=0

being a locally ﬁnite union of Z-sets, is a Z-set in U .

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31

1.5.4. Lemma. Suppose an ANR X = X1 ∪ X2 is the union of two open strongly C-universal subspaces X1 , X2 , where C is a closed-hereditary class. Then X is strongly C-universal. Proof. Fix U ∈ cov(X), C ∈ C, a closed subset B ⊂ C, and a map f : C → X that restricts to a Z-embedding on B. ¯1 ⊂ U ′ ⊂ Pick open sets U1 , U1′ , U2 , U2′ , U2′′ in X such that U1 ∪ U2 = X, U 1 ′ ′ ′ ′′ ′′ ¯ ⊂ X1 , and U ¯2 ⊂ U ⊂ U ¯ ⊂U ⊂U ¯ ⊂ X2 . Let V ∈ cov(X) be a cover U 1 2 2 2 2 ¯1 , V) ⊂ U ′ , St (X\U1 , V) ⊂ U2 . such that St V ≺ U, St (U 1 −1 ¯ ¯ ′ ). Since X is an ANR, there is a Let C1 = f (U1 ) and C1′ = f −1 (U 1 cover W ∈ cov(X) such that every map g : C1′ ∪B → X, W-close to f |C1′ ∪B extends to a map g¯ : C → X, V-close to f . Using strong C-universality of X1 , ﬁnd a Z-embedding g : C1′ → X1 such that g|C1′ ∩ B = f |C1′ ∩ B and (g, f |C1′ ) ≺ W. By the choice of W, the map g ∪ f |B : C1′ ∪ B → X extends to a map f1 : C → X with (f1 , f ) ≺ V. As an exercise we propose to the reader to verify that the map f1 |C1 ∪B : C1 ∪ B → X is a Z-embedding. (It should be noticed that generally, the map f1 |C1′ ∪ B can be non-injective). ¯ ′ ). Notice that Let C2 = f1−1 (U 2 ¯1 , V) ⊂ U2 , f1 (C\C1 ) ⊂ St (f (C\C1 ), V) ⊂ St (X\U so C\C1 ⊂ C2 and C1 ∪ C2 = C. Moreover, ClX (f1 (C\C1 )) ∩ ClX (f1 (C\C2 )) = ∅. Let V ′ ∈ cov(X) be a cover such that V ′ ≺ V, St (ClX (f1 (C\C1 )), V ′ ) ∩ ¯ ′ , V ′ ) ⊂ U ′′ . Using strong C-universality ClX (f1 (C\C2 )) = ∅, and St (U 2 2 of X2 , ﬁnd a Z-embedding f2 : C2 → X2 such that f2 |C2 ∩ (C1 ∪ B) = f1 |C2 ∩ (C1 ∪ B), and (f2 , f1 |C2 ) ≺ V ′ . Deﬁne ﬁnally a map f¯ : C → X by the formula { f1 (c), if c ∈ C1 ; ¯ f (c) = f2 (c), if c ∈ C2 . It can be shown that f¯ : C → X is a Z-embedding extending f |B and U-close to f . ∪ 1.5.5. Lemma. If an ANR X is the union X = i∈I Xi of a disjoint collection of open strongly C-universal subspaces, where C is a closed-hereditary class, then X is strongly C-universal. Proof. Given a cover U ∈ cov(X), a space C ∈ C, and a map f : C → X that restricts to a Z-embedding on a pregiven closed ∪ subset B ⊂ C, let Ci = f −1 (Xi ), i ∈ I. Notice ﬁrst, that each Xi = X\ j̸=i Xj is a closed set in X, hence each Ci is closed in C ∈ C, and consequently, Ci ∈ C.

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Using strong C-universality of Xi ’s, for each i ∈ I ﬁnd a Z-embedding f¯i : C that f¯i |B ∩∪Ci = f |B ∩ Ci and (f¯i , f |Ci ) ≺ U. Then ∪i → ¯Xi such ∪ ¯ f = i∈I fi : C = i∈I Ci → i∈I Xi = X is the required Z-embedding U-near to f and extending f |B. It turns out that the condition on C to be closed-hereditary in Proposition 1.5.1 is superﬂuous. Given a space C let F0 (C) denote the class of spaces homeomorphic to closed subsets of C. Evidently, F0 (C) is closed-hereditary. 1.5.6. Proposition. Suppose X is a strongly C-universal ANR, where C is a space. Then every open subspace U of X is strongly F0 (C)-universal. Proof. We will prove ﬁrstly this proposition for open sets U ⊂ X, contractible in X. (A set U ⊂ X is deﬁned to be contractible in X if there is a homotopy h : U × I → X such that h0 = id|U and h1 ≡ const). Remark that if U is contractible in X ∈ ANR then for every space C and its closed subset D ⊂ C, every map f : D → U extends to a map f¯ : C → X. Using this observation and the technique of the proof of Lemma 1.5.3, show that strong C-universality of X implies strong F0 (C)-universality of each open contractible in X subspace U ⊂ X. Since any open set U ⊂ X admits a cover consisting of open sets contractible in X, our statement follows from Proposition 1.5.1. A class of spaces C is deﬁned to be open-hereditary if every open subspace of each C ∈ C belongs to C. 1.5.7. Proposition. Suppose C is an open-hereditary class and X is an ANR such that each Z-set in X is strong. The space X is strongly Cuniversal if and only if for each open set U ⊂ X every map f : C → U , where C ∈ C, can be approximated by Z-embeddings. Proof. The “only if” part follows from Lemma 1.5.3. To prove the “if” part ﬁx a map f : C → X, C ∈ C, that restricts to a Z-embedding on a closed set B ⊂ C. By Corollary 1.4.7, we can approximate f by a map g : C → X such that g|B = f |B, g(C\B) ∩ f (B) = ∅, and so that g is closed over g(B) = f (B). Apply the hypothesis to the map g|C\B : C\B → X\g(B) to produce a Z-embedding g ′ : C\B → X\g(B). If g ′ is suﬃciently close to g|C\B, then the map f¯ : C → X deﬁned by f¯|B = f |B, f¯|C\B = g ′ , is a Z-embedding close to f with f¯|B = f |B. Given a space X we denote by SU (X) the class of spaces C such that X is strongly C-universal. By the other words, SU (X) is the largest class C such that X is strongly C-universal. Our aim now is to investigate relationship between properties of a space X and properties of the class SU (X).

1.5. STRONG UNIVERSALITY

33

A class of spaces C is deﬁned to be • topological if every topological copy of each C ∈ C belongs to C; • closed-additive if a space X belongs to C whenever it can be expressed as X = X1 ∪ X2 , where X1 , X2 ∈ C and one of X1 , X2 is closed in X; • local if a space X belongs to C iﬀ every point x ∈ X has a neighborhood U ∈ C. By σC (sometimes by ∪ σ-C or Cσ ) we denote the class of spaces X that can be expressed as X = n∈N Xn , where each Xn ∈ C is a closed subspace in X. 1.5.8. Theorem. Suppose X is an ANR. Then (1) (2) (3) (4)

the class SU (X) is topological and closed-hereditary; if every Z-set in X is strong then SU (X) is closed-additive; if X has SDAP then SU (X) is a local class; if X is a strong Zσ -space then SU (X) = σ-SU (X).

Proof. The ﬁrst statement of the theorem obviously follows from Proposition 1.5.6. The proofs of the second and the third statements depend on the following 1.5.9. Lemma. Let X be a∪strongly C-universal ANR and C be a space that can be expressed as C = n≥0 Cn , where C0 = ∅ and each Cn ∈ C is a closed subset in C with Cn ⊂ Cn+1 . Suppose f : C → X is a map such that the collection {f (C\Cn )}n≥0 is locally ﬁnite in X. Then for every cover U ∈ cov(X) there exists a Z-embedding f¯ : C → X such that (f¯, f ) ≺ U. Proof. Fix a cover U ∈ cov(X). By Ex.3 to §1.1.A, there is a cover V0 ∈ cov(X), V0 ≺ U, such that the collection {St (f (C\Cn ), V0 )}n≥0 is locally ﬁnite in X. Pick a sequence of covers {Vn }n≥1 ⊂ cov(X) so that St Vn ≺ Vn−1 and mesh Vn < 2−n for every n ∈ N. Let f0 = f and for every n ≥ 1, using strong Cn -universality of X, construct inductively a map fn : C → X such that fn |Cn : Cn → X is a Z-embedding, fn |Cn−1 = fn−1 |Cn−1 , and (fn , fn−1 ) ≺ Vn . Then the limit map f¯ = limn→∞ fn : C → X is V0 -close to f0 = f , and hence, the collection {f¯(C\Cn )}n≥0 is locally ﬁnite in X. It follows from the construction that f¯ is injective and f¯|Cn : Cn → X is a Z-embedding for every n ∈ N. By Ex.6 to §1.4, f¯ is a Z-embedding. This lemma implies the following useful

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1.5.10. Proposition. Let f : C → X be a map of a space C into a strongly C-universal ANR X. For every open set U ⊂ X and every cover U ∈ cov(U ) there is a Z-embedding g : f −1 (U ) → U such that (g, f |f −1 (U )) ≺ U. Now we prove the statement 2 of 1.5.8. Suppose every Z-set in X is strong and C is a space that can be expressed as C = C1 ∪ C2 , where C1 , C2 ∈ SU (X) and C1 is closed in C. To show that X is strongly Cuniversal, ﬁx a cover U ∈ cov(X), a closed subset B ⊂ C, and a map f : C → X that restricts to a Z-embedding on B. According to Corollary 1.4.7, we may assume that f (C\B) ∩ f (B) = ∅ and f is closed over f (B). Let V ∈ cov(X) be a cover with St 2 V ≺ U. Since X is an ANR, there is a cover W ∈ cov(X) such that every map f1 : C1 ∪ B → X, W-close to f |C1 ∪ B, extends to a V-close to f map f¯1 : C → X. By 1.5.10, there is a Z-embedding g : C1 \B → X\f (B) such that (g, f |C1 \B) ≺ W and d(g(c), f (c)) < 12 d(f (c), f (B)) for each c ∈ C1 \B. Then the map f1 : C1 ∪ B → X deﬁned by { f (c), if c ∈ B; f1 (c) = g(c), if c ∈ C1 \B is a Z-embedding W-close to f |C1 ∪ B. Extend f1 to a map f¯1 : C → X such that (f¯1 , f ) ≺ V. By 1.4.7, we can ﬁnd a map f˜1 : C → X such that (f˜1 , f¯1 ) ≺ V, f˜1 |C1 ∪ B = f¯1 |C1 ∪ B, f˜1 (C\(C1 ∪ B)) ∩ f¯1 (C1 ∪ B) = ∅ and f˜1 is closed over f˜1 (C1 ∪ B). Using 1.5.10, ﬁnd a Z-embedding h : C2 \(C1 ∪ B) → X\f˜1 (C1 ∪ B) such that (h, f˜1 |C2 \(C1 ∪ B)) ≺ V and d(h(c), f˜1 (c)) < 1 ˜ ˜ ¯: C → X d( f (c), f (C ∪ B)) for each c ∈ C \(C ∪ B). Then the map f 1 1 1 2 1 2 deﬁned by the formula { ˜ f1 (c), if c ∈ C1 ∪ B; ¯ f (c) = h(c), if c ∈ C2 \(C1 ∪ B) is a Z-embedding U-close to f and extending f |B. The second statement of the theorem is proved. To prove the third statement, suppose X has SDAP and C is a space such that each point c ∈ C has a neighborhood U ∈ SU (X). Fix a cover U ∈ cov(X), a closed set B ⊂ C, and a map f : C → X that restricts to a Z-embedding on B. By Proposition 1.4.3, each Z-set in X is strong. Thus, according to Corollary 1.4.7, we may assume that f (C\B) ∩ f (B) = ∅ and f is closed over f (B). Let C ′ = C\B and V ∈ cov(X\f (B)) be a cover such that St V ≺ U and St V ≺ {B(x, d(x, f (B))/2) | x ∈ X\f (B)}. By the ﬁrst statement of the theorem, the class SU (X) is closed-hereditary. Using this fact and the fact that each point c ∈ C has a neighborhood

1.5. STRONG UNIVERSALITY

35

U ∈ SU (X), pick ∪ a countable collection {Fn }n∈N of closed subsets of C such that C ′ = n∈N Int Fn and each Fn ∈ SU (X). Let Cn = F1 ∪ · · · ∪ Fn , n ∈ N. By the second statement of the theorem, each Cn ∈ SU (X). Notice also that the collection {C ′ \Cn }n∈N is locally ﬁnite in C ′ . By 1.3.3, the open subspace X\f (B) of X has SDAP. Then by 1.3.4, there is a map f ′ : C ′ → X\f (B) such that (f ′ , f |C ′ ) ≺ V and the collection {f ′ (C ′ \Cn )}n∈N is locally ﬁnite in X\f (B). Using 1.5.10, ﬁnd a Zembedding f˜ : C ′ → X\f (B) such that (f˜, f ′ ) ≺ V. Then the map f¯ : C → X deﬁned by the formula { f¯(c) =

f (c), if c ∈ C f˜(c), if c ∈ C\B

is a Z-embedding U-close to f and extending f |B. ∪ Suppose ﬁnally X is a strong Zσ -space, and C ∈ σ-SU (X), i.e., C = i∈N Ci , where each Ci ∈ SU (X) is a closed set in C. Fix a cover U0 ∈ cov(X), a closed subset B ⊂ C, and a map f : C → X that restricts to a Z-embedding on B. By Theorem 1.4.10, X satisﬁes SDAP, and hence, as we have already proved, the class SU (X) is local. This yields that every open subspace of each A ∈ SU (X) belongs to SU (X). In particular, for each map g : C → X the restriction g|Ci \Ci−1 can be approximated by Z-embeddings. Using this fact, we construct a sequence {fi : C → X} satisfying fi |Ci is a Z-embedding, fi |Ci−1 = fi−1 |Ci−1 , fi is a closed map over fi (Ci ) and fi (C\Ci ) ∩ fi (Ci ) = ∅, d(fi (c), fi−1 (c)) ≤ 12 d(fi−1 (c), fi−1 (Ci−1 )) for c ∈ C\Ci−1 , ClX fi (C) ∩ (Xi \fi−1 (Ci−1 )) = ∅, d(fi (c), fi−1 (c)) < 14 d(fi−1 (c), Xi−1 ) for c ∈ C\Ci−1 , and (fi , fi−1 ) ≺ Ui for a sequence {Ui ∈ cov(X)} with mesh Ui < 2−i and St Ui ≺ Ui−1 . As usual, f0 = f . Suppose fi−1 has been constructed, and consider fi−1 |Ci \Ci−1 : Ci \Ci−1 → X\fi−1 (Ci−1 ). Approximate this map by a Zembedding h : Ci \Ci−1 → X\fi−1 (Ci−1 ) so close to fi−1 |Ci \Ci−1 that h ex˜ : C\Ci−1 → X\fi−1 (Ci−1 ) and h ˜ is close to fi−1 |C\Ci−1 . tends to a map h Note that each Z-set in X\fi−1 (Ci−1 ) is strong, and hence by Corollary 1.4.7, ˜ ′ : C\Ci−1 → X\fi−1 (Ci−1 ) close to h ˜ with h ˜ ′ |Ci \Ci−1 = there is a map h ′ ′ ˜ ˜ ˜ ˜ ˜ h|Ci \Ci−1 , h (C\Ci ) ∩ h(Ci \Ci−1 ) = ∅, and h is closed over h(Ci \Ci−1 ). ˜ ′ . Then Deﬁne fi : C → X by fi |Ci−1 = fi−1 |Ci−1 , fi |C\Ci−1 = h f ′ = lim fi : C → X i→∞

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is a closed embedding St U1 -close to f . Note that f ′ may not be a Zembedding (f ′ can even be a homeomorphism!). To get a Z-embedding, ﬁx a countable set A0 = {α01 , α02 , . . . } dense in C(Q, X). Along with the sequence {fi : C → X} we construct countable dense sets Ai = {αi1 , αi2 , . . . } ⊂ C(Q, X) so that αik (Q) ∩ fi (Ci ) = ∅ for k ≤ i, k for k ≤ i − 1, and αik = αi−1 k k ) < 2−k for k ≥ i. d(αi , αi−1 This ∪ construction is possible because compact subsets of X (in particular, k≤i αik (Q)) are (strong) Z-sets in X (see Proposition 1.4.9). Then ∪∞ {α11 , α22 , . . . } ⊂ C(Q, X) is a dense set, and f ′ (C) ∩ ( k=1 αkk (Q)) = ∅. It follows that f ′ (C) is a Z-set in X. 1.5.11. Theorem. If X is a strongly C-universal ANR, where C is a space and Y is an ANR then X × Y is strongly C-universal, provided every Z-set in X × Y is strong. Proof. Fix a cover U ∈ cov(X × Y ), a closed subset B ⊂ C, and a map f : C → X that restricts to a Z-embedding on B. Suppose that each Z-set in X × Y is strong. Then, according to Corollary 1.4.7, we may assume that f (C\B) ∩ f (B) = ∅ and f is closed over f (B). Fix metrics dX and dY on X and Y respectively, and consider on X × Y the metric d deﬁned by d((x, y), (x′ , y ′ )) = max{dX (x, x′ ), dY (y, y ′ )}. Let ε : X × Y → (0, 1] be a function such that {B(x, ε(x))}x∈X×Y ≺ U, and deﬁne a map δ : X × Y → [0, 1] letting δ(x) =

1 min{ε(x), d(x, f (B))}. 2

Let C0 = ∅ and Cn = (δ ◦ ∪ f )−1 ([2−n , 1]) for n ≥ 1. Evidently, each Cn is closed in C and C\B = n≥0 Cn . Denote by pX : X × Y → X, pY : X × Y → Y the natural projections. Using strong C-universality of X, construct inductively a map g : C\B → X such that for every n ∈ N the following conditions are satisﬁed: g|Cn : Cn → X is a Z-embedding and dX (g(c), pX ◦ f (c)) < 2−n for each c ∈ Cn \Cn−1 . Then the map f¯ : C → X × Y deﬁned by the formula { f (c), if c ∈ B ¯ f (c) = (g(c), pY ◦ f (c)), if c ∈ C\B is a Z-embedding extending f |B and U-close to f .

1.5. STRONG UNIVERSALITY

37

Exercises and Problems to §1.5. 1. Let X be an AR and ∗ ∈ X any point. a) Show that X ω is a strongly universal AR. b) Show that W (X, ∗) = {(xn ) ∈ X ω | xn ̸= ∗ for ﬁnitely many of n’s} is a strongly universal AR. c) Show that each space Y with W (X, ∗) ⊂ Y ⊂ X ω is a strongly F0 (X)-universal AR. Hint: see the section 4.1. 2. (Enlarging Theorem). Let X be an ANR, and A a strong Z-set in X. Suppose X\A is a strongly C-universal space for a class C of spaces. Show that the space X is strongly C-universal. 3. Find an AR X and a point ∗ ∈ X such that {∗} is a Z-set in X, X\{∗} is strongly M1 -universal (recall that M1 stands for the class of all Polish spaces), but X is not strongly M1 -universal. 4. (Deleting Theorem). Let X be a strongly C-universal ANR such that every Z-set in ¯ is a Z-set in X. Show that X is strong, and let A be a subset of X whose closure A X\A is a strongly C-universal space. 5. (C.Bessaga, A.Pelczy´ nski [1975]). Let X be an ANR and K a class of spaces. Suppose X contains a K-skeleton that is a tower of Z-sets X1 ⊂ X2 ⊂ · · · ⊂ Xn ⊂ · · · ⊂ X, satisfying the conditions i) each Xn belongs to K;

ii) given a cover U ∈ cov(X), a space K ∈ K, a closed subset B ⊂ K, n ∈ N, and a map f : K → X such that f |B : B → X is a closed embedding with f (B) ⊂ Xn , there exist an m ∈ N and a closed embedding f¯ : K → Xm such that (f¯, f ) ≺ U and f¯|B = f |B. Show that a) each space K ∈ K is compact; b) the space X is strongly K-universal. 6. Let X be an ANR with LCAP and C a class of spaces. Suppose X contains a tower X1 ⊂ X2 ⊂ · · · ⊂ Xn ⊂ . . . of Z-sets in X such that i) each Xn is a strongly C-universal ANR; ii) given a cover U ∈ cov(X), n ∈ N, and a map f : K → X of a ﬁnite-dimensional compactum, there exist an m ∈ N and a map f¯ : K → Xm such that (f¯, f ) ≺ U and f¯|f −1 (Xn ) = f |f −1 (Xn ). Show that the space X is strongly C-universal. Hint: See T.Banakh [199?b]. 7. Let X be a subset of a linear metric space ∪ and X1 ⊂ X2 ⊂ · · · ⊂ Xn ⊂ . . . be a tower of convex subsets of X such that n∈N Xn is dense in X. Show that this tower satisﬁes the condition (ii) of Problem 6. 8. (Open Problem) Does there exists a space X with the properties: (i) X is an ANR with SDAP; (ii) X is strongly Z(X)-universal, where Z(X) denotes the collection of Z-sets in X; (iii) X is not strongly F0 (X)-universal.

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9. Give an example of an ANR X that is strongly Z(X)-universal, but not strongly F0 (X)-universal. 10. (Open Problem) Let X, Y be two strongly universal ANR’s. Is the product X × Y strongly universal? More generally, suppose X is a strongly C-universal ANR and Y is a strongly D-universal ANR. Is the product X × Y strongly C × D-universal? 11. Show that each strongly ⊕n∈N I n -universal space satisﬁes SDAP.

§1.6. Absorbing and coabsorbing spaces. The following are the central conceptions of the book. 1.6.1. Definition. Let C be a class of spaces. A space X is deﬁned to be C-absorbing if a) X is a strongly C-universal ANR satisfying SDAP; b) X ∈ σC; c) X is a Zσ -space. Obviously, every C-absorbing space is of the ﬁrst Baire category. Next, we deﬁne C-coabsorbing spaces that are Baire spaces. 1.6.2. Definition. Let C be a class of spaces. A space X is deﬁned to be C-coabsorbing if a) X is a strongly C-universal ANR satisfying SDAP; b) every Z-set of X belongs to the class C; c) X is a co-Zσ -space, that is X contains a homotopy dense absolute Gδ -subset. Notice that a space is M1 -coabsorbing if and only if it is an s-manifold (just apply Characterization Theorem 1.1.14). Similarly to s-manifolds, topological classiﬁcation of C-absorbing and C-coabsorbing spaces coincides with homotopical one. Namely, we have the following two fundamental facts. 1.6.3. Theorem. Let C be a class of spaces. Two C-absorbing spaces are homeomorphic if and only if they are homotopy equivalent. 1.6.4. Theorem. Let C be a class of spaces. Two C-coabsorbing spaces are homeomorphic if and only if they are homotopy equivalent. Proofs of these theorems are based on the following lemma generalizing Z-Set Unknotting Theorem for s-manifolds. 1.6.5. Lemma. Let M be an s-manifold. For every cover U ∈ cov(M ) there exists a cover V ∈ cov(M ) such that for every two subsets A, B ⊂ M ¯ B ¯ are Z-sets in M , every homeomorphism h : A → B, whose closures A, ¯ : A ∪ (M \A) ¯ → V-close to id, can be extended to a homeomorphism h ¯ B ∪ (M \B), U-close to id.

1.6. ABSORBING AND COABSORBING SPACES

39

Proof. Given a cover U ∈ cov(M ) let U ′ ∈ cov(M ) be a cover with St U ′ ≺ U and U ′′ ∈ cov(M ) a cover such that any two U ′′ -near maps into M are U ′ homotopic. Let V ∈ cov(M ) be a cover with St 2 V ≺ U ′′ . We are going to show that V is as required. ¯ B ¯ are Z-sets in M . Suppose Let A, B be two subsets in M such that A, h : A → B is a homeomorphism V-close to id|A. By Lavrentiev Theorem ˜ : A˜ → B ˜ this homeomorphism can be extended to a homeomorphism h ˜ ¯ ˜ ¯ ¯ ˜ ¯ ˜ between certain Gδ -sets A ⊂ A and B ⊂ B. Notice that A\A, B\B are Zσ -sets in M . Thus, using Strong Negligibility Theorem 1.1.19, we can ﬁnd ¯ A) ˜ → M and hB : M \(B\ ¯ B) ˜ → M , V-close homeomorphisms hA : M \(A\ ˜ ¯ A) ˜ and B ˜ to the respective identity embeddings. Since A is a Z-set in M \(A\ ¯ B), ˜ hA (A) ˜ and hB (B) ˜ are Z-sets in M . Hence the map is a Z-set in M \(B\ ˜ ◦ h−1 |hA (A) ˜ : hA (A) ˜ → hB (B) ˜ ⊂ M is a Z-embedding. It is easy f = hB ◦ h A ˜ are U ′ -homotopic. to see that (f, id) ≺ St 2 V ≺ U ′′ and thus f and id|hA (A) By Z-Set Unknotting Theorem 1.1.18, there is a homeomorphism f¯ : M → ¯ = h−1 ◦ f¯◦ hA |A ∪ M extending f , St (U ′ , V)-homotopic to id. Finally let h B ¯ ¯ id) ≺ ¯ ¯ ¯ (M \A) : A ∪ (M \A) → B ∪ (M \B) and notice that h|A = h and (h, ′ 2 ′ ′′ ′ St (U , St V) ≺ St (U , U ) ≺ St U ≺ U. In fact, we need the following minor modiﬁcation of Lemma 1.6.5. 1.6.6. Lemma. Let M , N be s-manifolds and f : M → N a homeomorphism. For every cover U ∈ cov(N ) there exists a cover V ∈ cov(N ) such ¯ B ¯ are Z-sets that for every two subsets A ⊂ M , B ⊂ N whose closures A, in M , N respectively, every homeomorphism h : A → B, V-close to f |A, ¯ : A ∪ (M \A) ¯ → B ∪ (M \B), ¯ U-close can be extended to a homeomorphism h to f . Proof of Theorem 1.6.3. Let X, Y be two homotopically equivalent Cabsorbing spaces. According to 1.5.8, we can assume the class C to be topological, closed-hereditary, closed-additive, and local. The space X, being an ANR with SDAP, admits a homotopy dense embedding into an s-manifold M . Analogously, for the space Y there exists an s-manifold N containing Y as a homotopy dense subset. Obviously, M is homotopically equivalent to X and N is homotopically equivalent to Y . Consequently, the s-manifolds M and N have the same homotopy type. Then by Classiﬁcation Theorem 1.1.15, there exists a homeomorphism f0 : M → N . Fix complete metrics ρ and d on M and N respectively. Now we ∪ will construct a∪ homeomorphism f : X → Y close to f0 |X. ∞ ∞ Write X = n=1 Xn , Y = n=1 Yn , where each Xn ⊂ Xn+1 ∈ C is a Z-set in X, and Yn ⊂ Yn+1 ∈ C is a Z-set in Y . Let X0 = Y0 = ∅. Fix a cover U0 ∈ cov(N ), and let {Un }n≥1 ⊂ cov(N ), {Vn }n≥0 ⊂ cov(M ) be sequences of covers such that St Un ≺ Un−1 , St Vn ≺ Vn−1 , mesh Un < 2−n , and mesh Vn < 2−n for n ≥ 1.

40

BASIC THEORY

To ﬁnd a homeomorphism f : X → Y we will construct inductively sequences of homeomorphisms {gn : Nn → Mn }, {fn : Mn′ → Nn′ }, where Mn , Mn′ ⊂ M and Nn , Nn′ ⊂ N are subsets with X ⊂ Mn ∩ Mn′ and Y ⊂ Nn ∩ Nn′ such that the following conditions are satisﬁed for every n: (1n ) (2n ) (3n ) (4n ) (5n ) (6n ) (7n ) (8n ) (9n ) (10n )

Nn = Yn ∪ fn (Xn ) ∪ N \Yn ∪ fn (Xn ); Mn = gn (Yn ) ∪ Xn ∪ M \gn (Yn ) ∪ Xn ; Mn′ = Xn ∪ gn−1 (Yn−1 ) ∪ M \Xn ∪ gn−1 (Yn−1 ); Nn′ = fn (Xn ) ∪ Yn−1 ∪ N \fn (Xn ) ∪ Yn−1 ; −1 −1 −1 (fn , gn−1 ) ≺ Un+1 , (fn , gn−1 ) ≺ gn−1 (Vn+1 ); −1 −1 −1 (gn , fn ) ≺ Vn+1 , (gn , fn ) ≺ fn (Un+1 ); −1 fn |Xn−1 ∪ gn−1 (Yn−1 ) = gn−1 |Xn−1 ∪ gn−1 (Yn−1 ); −1 gn |fn (Xn ) ∪ Yn−1 = fn |fn (Xn ) ∪ Yn−1 ; fn (Xn ) is a Z-set in Y ; gn (Yn ) is a Z-set in X.

Then the maps f = limn→∞ fn |X and g = limn→∞ gn |Y are well-deﬁned and continuous, and f ◦ g = id|Y , g ◦ f = id|X. Letting M0 = M0′ = M , N0 = N0′ = N , and g0 = f0−1 we proceed inductively. Assume that fi , gi (satisfying (1i )—(10i ) for i = 1, . . . , n) have been constructed. We will construct a map fn+1 . It follows from (1n ) and ◦

◦

(2n ) that gn (N n ) = M n , where ◦

N n = N \Yn ∪ fn (Xn )

◦

and M n = M \gn (Yn ) ∪ Xn ◦

are open subsets in N and M respectively. Pick a cover U ∈ cov(N n ) such that U ≺ Un+2 , U ≺ gn−1 (Vn+2 ), and ◦

◦

U ≺ {B(y, d(y, N \N n )/3) | y ∈ N n }.

(1) ◦

Let W ∈ cov(N n ) be a cover satisfying the conditions of Lemma 1.6.6 ◦

◦

with respect to the cover U and the homeomorphism gn−1 : M n → N n . Since ◦

◦

the class C is local, Xn+1 ∩ M n ∈ C. By 1.5.6, the space Y ∩ N n is strongly ◦

◦

C-universal. Hence, there is a Z-embedding e : Xn+1 ∩ M n → Y ∩ N n , W-close to gn−1 . By the choice of W, this embedding can be extended to a homeomorphism ◦

◦

◦

◦

◦

¯ n+1 ) → e(Xn+1 ∩ M n ) ∪ N n \e(Xn+1 ∩ M n ) e¯ : (Xn+1 ∩ M n ) ∪ (M n \X such that (2)

(¯ e, gn−1 ) ≺ U.

1.6. ABSORBING AND COABSORBING SPACES

41

′ ′ Let Mn+1 = Xn+1 ∪ gn (Yn ) ∪ M \Xn+1 ∪ gn (Yn ), and notice that Mn+1 = ◦ ◦ ¯ Xn ∪ gn (Yn ) ∪ (Xn+1 ∩ M n ) ∪ (M n \Xn+1 ). It follows from (1) and (2) that ′ the map fn+1 : Mn+1 → N deﬁned by the formula

{ fn+1 (x) =

gn−1 (x), if x ∈ Xn ∪ gn (Yn ), e¯(x),

◦

◦

¯ n+1 ) if x ∈ (Xn+1 ∩ M n ) ∪ (M n \X

is an embedding satisfying (5n+1 ), (7n+1 ), and (9n+1 ). One can easily verify ′ ′ that the set Nn+1 = fn+1 (Mn+1 ) satisﬁes the condition (4n+1 ). Using the same arguments, construct a homeomorphism gn+1 : Nn+1 → Mn+1 to satisfy the conditions (1n+1 ), (2n+1 ), (6n+1 ), (8n+1 ), and (10n+1 ). Proof of Theorem 1.6.4. Let X, Y be two homotopically equivalent Ccoabsorbing spaces. According to 1.5.8, we can assume the class C to be topological, closed-hereditary, closed-additive and local. Without loss of generality, the class C contains a non-complete-metrizable space (in the other case, X and Y , being complete-metrizable ANR with SDAP, are smanifolds). Then, the spaces X, Y , being strongly C-universal, are nowhere topologically complete. By Theorem 1.3.2, the space X admits a homotopy dense embedding into ˜ . Let GX ⊂ X be a homotopy dense absolute Gδ -subset in an s-manifold M ∪∞ ˜ for ˜ X. Write M \GX = n=1 Fn , where Fn ⊂ Fn+1 are closed subsets in M ∪∞ ˜ ¯n) every n. Let Xn = Fn ∩ X, n ∈ N, and remark that M = M \ n=1 (Fn \X ˜ containing X. By Strong Negligibility is a homotopy dense Gδ -subset in M Theorem 1.1.19, M is an s-manifold. ∪∞ ∪∞ ¯ Remark that M = GX ∪ n=1 X n , X = GX ∪ n=1 Xn , and each Xn is a Z-set ∪∞ in X, and hence Xn ∈ C. Since X is nowhere topologically complete, n=1 Xn is dense in X. Analogously, for the C-coabsorbing space Y we ﬁnd an s-manifold N containing Y as a homotopy ∪∞ ¯ dense subset so that we can write Y = GY ∪ ∪∞ dense absolute Gδ Y , N = G ∪ Y n=1 Yn , where GY is a homotopy n=1 n ∪∞ set in Y , Yn ’s are Z-sets in Y with Y¯n ∩ GY = ∅, and n=1 Yn being dense in Y . Since the s-manifolds M , N are homotopy equivalent, by Classiﬁcation Theorem 1.1.15, there is a homeomorphism f0 : M → N . Similarly as in the proof of 1.6.3, we will construct a homeomorphism f : X → Y close to f0 |X. The idea of constructing f is the same with the only diﬀerence that we should impose more strict control on convergence of fn ’s and gn ’s to guarantee f (GX ) ⊂ Y and g(GY ) ⊂ X. Fix a cover U0 ∈ cov(N ), and complete metrics ρ and d on M and N respectively. Let g0 = f −1 , V0 = f0−1 U0 , and X0 = Y0 = ∅.

42

BASIC THEORY

For every n ∈ N we shall construct inductively homeomorphisms gn : ◦

◦

◦

◦

Nn → Mn , fn : Mn′ → Nn′ , open sets M n , M ′n ⊂ M , N n , N ′n ⊂ N , and ◦

◦

covers Un ∈ cov(N n ), Vn′ ∈ cov(M ′n ) such that the following conditions are satisﬁed: (1n ) Nn = Yn ∪ fn (Xn ) ∪ N \Yn ∪ fn (Xn ); ◦

(2n ) N n = N \Yn ∪ fn (Xn ); (3n ) Mn = gn (Yn ) ∪ Xn ∪ M \gn (Yn ) ∪ Xn ; ◦

(4n ) M n = M \gn (Yn ) ∪ Xn ; (5n ) Mn′ = Xn ∪ gn−1 (Yn−1 ) ∪ M \Xn ∪ gn−1 (Yn−1 ); ◦

(6n ) M ′n = M \Xn ∪ gn−1 (Yn−1 ); (7n ) Nn′ = fn (Xn ) ∪ Yn−1 ∪ N \fn (Xn ) ∪ Yn−1 ; ◦

(8n ) N ′n = N \fn (Xn ) ∪ Yn−1 ; (9n ) Un ≺ fn Vn′ , St 2 Un ≺ gn−1 Vn′ , mesh Un < 2−n , ◦

◦

Un ≺ {B(y, d(y, N \N n ) | y ∈ N n }; (10n ) Vn′ ≺ gn−1 Un−1 , St 2 Vn′ ≺ fn−1 Un−1 , mesh Vn′ < 2−n , ◦

(11n ) (12n ) (13n ) (14n ) (15n ) (16n )

◦

Vn′ ≺ {B(x, ρ(x, M \M ′n ) | x ∈ M ′n }; −1 (fn , gn−1 ) ≺ Un−1 ; (gn , fn−1 ) ≺ Vn′ ; −1 fn |Xn−1 ∪ gn−1 (Yn−1 ) = gn−1 |Xn−1 ∪ gn−1 (Yn−1 ); gn |fn (Xn ) ∪ Yn−1 = fn−1 |fn (Xn ) ∪ Yn−1 ; fn (Xn ) is a Z-set in Y ; gn (Yn ) is a Z-set in X.

Inductive construction. Assume that fi , gi , Ui , Vi′ (satisfying (1i )— (16i ) for i = 1, . . . , n − 1) have been constructed. Repeating the arguments from the proof of Theorem 1.6.3, construct a homeomorphism fn : Mn′ → Nn′ satisfying the conditions (5n ), (7n ), (11n ), (13n ), and (15n ). Deﬁne ◦

◦

the sets M ′n and N ′n by formulae (6n ) and (8n ), and pick up any cover ◦

Vn′ ∈ cov(M ′n ) that satisﬁes the condition (10n ). Next, construct a homeomorphism gn : Nn → Mn to satisfy the conditions (1n ), (3n ), (12n ), (14n ), ◦

◦

and (16n ). Deﬁne sets N n , M n by formulae (2n ), (4n ). Finally, let Un ∈ ◦

cov(M n ) be any cover satisfying (9n ). The inductive step is over. It follows from (9n )—(12n ) that the maps f = limn→∞ fn |X and g = limn→∞ gn |Y are well-deﬁned and continuous. Let us show ∪∞that f (X) ⊂ Y .∪ Notice ﬁrstly that ∪ by (13n )–(16n ), n ∈ N, we get ∪ f ( n=1 Xn ) ⊂ Y , ∞ ∞ ∞ g( n=1 Yn ) ⊂ X, f ◦ g| n=1 Yn ∪ f (Xn ) = id, and g ◦ f | n=1 Xn ∪ g(Yn ) =

1.6. ABSORBING AND COABSORBING SPACES

43

id. Thus (1)

f(

∞ ∪

Xn ∪ g(Yn )) ⊂ Y.

n=1 ◦

◦

By the choice of the covers Un , Vn′ , n ∈ N, we have f (X ∩ M n ) ⊂ N n for every n. Consequently, f (X ∩

(2)

∞ ∩

◦

M n) ⊂

n=1

∞ ∩

◦

Nn

n=1

By (4n ), (16n ), n ∈ N, we have (3)

∞ ∩

X∩

◦

M n = X\

n=1

∞ ∪

gn (Yn ) ∪ Xn = X\

∞ ∪

g(Yn ) ∪ Xn .

n=1

n=1

In the meantime, (2n ), n ∈ N, imply (4)

∞ ∩ n=1

∞ ∪

◦

Nn = N\

Yn ∪ fn (Xn ) ⊂ N \

n=1

∞ ∪

Y¯n = GY ⊂ Y.

n=1

Summing up (1)–(4), we obtain that f (X) ⊂ Y . Thus the composition ∪∞ g ◦ f∪: X → M is well deﬁned and continuous. Since g ◦ f | n=1 Xn = id ∞ and n=1 Xn is dense in X, we get f ◦ g = id. By the same reasoning we can prove that g(Y ) ⊂ X and g ◦ f = id. We deﬁne a space X to be absorbing (resp. coabsorbing) if X is F0 (X)absorbing (resp. F0 (X)-coabsorbing), where F0 (X) stands for the class of spaces homeomorphic to closed subspaces of X. Theorems 1.6.3 and 1.6.4 can be reformulated as follows. 1.6.7. Theorem. Two (co-)absorbing spaces X and Y are homeomorphic iﬀ they are homotopy equivalent and F0 (X) = F0 (Y ). Exercises and Problems to §1.6. 1. Show that for a space X the following conditions are equivalent: a) X is C-absorbing for certain class C; b) X is absorbing; c) X is Z(X)-absorbing; 2. Show that an ANR space X with ∪ SDAP is K-absorbing, provided X contains a K∞ skeleton X1 ⊂ X2 ⊂ . . . with X = n=1 Xn .

44

BASIC THEORY

3. Using the previous exercise prove that the spaces σ, l2f are M0 (ω)-absorbing and the spaces Σ, Bd(Q) are M0 -absorbing. Here M0 (M0 (ω)) stands for the class of all (ﬁnite-dimensional) compacta. 4. Let C be a topological closed-hereditary closed-additive local class, and Ω be a C(co)absorbing AR. We deﬁne a space X to be an Ω-manifold if each point x of X has a neighborhood homeomorphic to an open subset of Ω. Prove the following fundamental results concerning Ω-manifolds. a) (Characterization Theorem). A space X is an Ω-manifold if and only if X is a C-(co)absorbing space; b) (Open Embedding Theorem). For every two Ω-manifolds M , N there exists an open embedding M → N ; c) (Triangulation Theorem). Suppose the class C is [0,1]-stable, that is C ×[0, 1] ∈ C for every C ∈ C. Every Ω-manifold M is homeomorphic to K × Ω, where K is any locally ﬁnite simlicial complex homotopically equivalent to M ; d) (Z-Set Unknotting Theorem.) Suppose A is a Z-set in an Ω-manifold M , U , V ∈ cov(M ) covers, and h : A → M a Z-embedding U -homotopic to id|A. Then there is a ¯ : M → M extending h and St (U , V)-homotopic to id|M . homeomorphism h 5. Suppose X a C-absorbing space for a [0, 1]-stable class C. Prove that both X × I and X × σ are homeomorphic to X. 6. Let X be an AR and A ⊂ X. Show that a) X ω is an absorbing space if and only if X ω is a Zσ -space; b) X ω is a coabsorbing space if and only if X ω is a non-compact co-Zσ -space; ¯ ̸= X then W (X, A) is an absorbing space; c) if A d) if A is a Zσ -set in X then W (X, A) is absorbing; e) if A contains a homotopy dense in X absolute Gδ -set then W (X, A) is a coabsorbing space. 7. (Open Problem) Is the product of two (co)absorbing spaces a (co)absorbing space?

§1.7. Strongly universal and absorbing pairs. The results of the previous section show us the importance of the conceptions of a strongly universal space and of an absorbing space. In this section we generalize these conceptions to the case of pairs of spaces. Two fundamental results will be proved: Uniqueness Theorem for absorbing pairs, and a theorem revealing interplay between strongly universal spaces and pairs. Further, writing the symbol (M, X) we understand that X is a subspace of M . We call two pairs (M, X), (M ′ , X ′ ) homeomorphic if there exists a homeomorphism h : M → M ′ with h(X) = X ′ . Let (M, X), (K, C) be two pairs. The pair (M, X) is deﬁned to be strongly (K, C)-universal if for every cover U ∈ cov(M ), every closed subset B ⊂ K, and every map f : K → M whose restriction on B is a Z-embedding with (f |B)−1 (X) = B ∩ C there exists a Z-embedding f¯ : K → M such that f¯|B = f |B, (f¯, f ) ≺ U, and f¯−1 (X) = C. ⃗ We call a pair (M, X) strongly C-universal, where C⃗ is a class of pairs if

1.7. STRONGLY UNIVERSAL AND ABSORBING PAIRS

45

⃗ (M, X) is strongly (K, C)-universal for every (K, C) ∈ C. We deﬁne a class C⃗ of pairs to be

• topological if any pair (K ′ , C ′ ) homeomorphic to a pair (K, C) ∈ C⃗ be⃗ longs to the class C; • closed-hereditary if for every pair (K, C) ∈ C⃗ and every closed subset ⃗ B ⊂ K the pair (B, B ∩ C) belongs to the class C; • closed-additive, provided a pair (K, C) belongs to C⃗ whenever K can be ⃗ written as K1 ∪ K2 , where K1 is closed in K, and (Ki , Ki ∩ C) ∈ C, i = 1, 2; • local if a pair (K, C) belongs to C⃗ if and only if every point x ∈ K has a ⃗ neighborhood U ⊂ K such that (U, U ∩ C) ∈ C.

For a pair (M, X) denote by SU (M, X) the class of couples (K, C) such that (M, X) is strongly (K, C)-universal, and by F0 (M, X) the class of pairs homeomorphic to couples (F, F ∩ X), where F runs over closed subsets of M. For strongly universal pairs the following counterparts of 1.5.6, 1.5.1, 1.5.8, and 1.5.10 hold. 1.7.1. Proposition. Suppose M is an ANR and (M, X) is a strongly (K, C)-universal pair. Then for every open subset U ⊂ M the pair (U, U ∩ X) is strongly F0 (K, C)-universal. 1.7.2. Proposition. Let (M, X), (K, C) be pairs and M an ANR. The pair (M, X) is strongly (K, C)-universal if and only if every point x ∈ M has a neighborhood U ⊂ M such that the pair (U, U ∩ X) is strongly (K, C)universal. 1.7.3. Proposition. Let M be an ANR and X ⊂ M . a) The class SU (M, X) is topological and closed-hereditary; b) If every Z-set in M is strong then the class SU (M, X) is closed-additive. c) If M satisﬁes SDAP then SU (M, X) is a local class. 1.7.4. Proposition. Let M be an ANR, (M, X) a strongly (K, C)-universal pair, f : K → M a map, U ⊂ M an open set, and U ∈ cov(U ) a cover. Then there is a Z-embedding g : f −1 (U ) → U such that (g, f |f −1 (U )) ≺ U and g −1 (X) = C ∩ f −1 (U ). Proofs of 1.7.1—1.7.4 repeat the corresponding proofs from the section 1.5 and are left to the reader. 1.7.5. Proposition. Suppose (M, X) is a strongly (K, C)-universal pair, where M is an ANR such that every Z-set in M is strong. Then for every subset A ⊂ M whose closure A¯ is a Z-set in M the pair (M, X ∪ A) is strongly (K, C)-universal.

46

BASIC THEORY

Proof. Fix a cover U ∈ cov(M ), a closed subset B ⊂ K, and a map f : K → M such that f |B : B → M is a Z-embedding with (f |B)−1 (X ∪A) = B ∩C. Remark that D = f (B) ∪ A¯ is a strong Z-set in X. Then, by Lemma 1.4.7, without loss of generality, we can assume that f (K\B) ∩ D = ∅ and f is closed over D. Let V ∈ cov(M \D) be a cover such that V ≺ U and V ≺ {B(x, d(x, D)/2) | x ∈ M \D}. By Proposition 1.7.4, there is a Zembedding g : K\B → M \D V-close to f |K\B and such that g −1 (X) = C\B. Then the map f¯ : K → M deﬁned by { f¯(X) =

f (x), if x ∈ B; g(x),

if x ∈ K\B

is a Z-embedding with the properties (f¯, f ) ≺ U, f¯|B = f |B, and f¯−1 (X ∪ A) = C. The conception of an absorbing space also can be generalized to the case ⃗ of pairs. For a class of pairs C⃗ denote ∪∞ by σ C the class of pairs (K, C) such that K can be expressed as K = n=1 Kn , where for every n ≥ 1 Kn is a ⃗ closed set in K with (Kn , Kn ∩ C) ∈ C. ⃗ A pair (M, X) is deﬁned to be C-absorbing, where C⃗ is a class of pairs if ⃗ a) (M, X) is strongly C-universal; ⃗ b) there is a Zσ -set Z ⊂ M containing X so that (Z, X) ∈ σ C. Similarly as for absorbing spaces, we have the following Uniqueness The⃗ orem for C-absorbing couples. 1.7.6. Theorem. Let C⃗ be a class of pairs, M either a Q-manifold or an s⃗ manifold, and (M, X), (M, X ′ ) two C-absorbing pairs. Then for every cover U ∈ cov(M ) there is a homeomorphism f : M → M such that (f, id) ≺ U and f (X) = X ′ . Proof. Fix a cover U0 ∈ cov(M ) and a complete metric d on M . Let {Un }∞ n=1 ⊂ cov(M ) be a sequence of covers such that St Un ≺ Un−1 and mesh Un < 2−n for every n ∈ N. Let Z, Z ′ ⊂ M be Zσ -subsets in M such that X ⊂ Z, X ′ ⊂ Z ′ , and ⃗ By Theorem 1.7.3, we can assume the class C⃗ to (Z, X), (Z ′ , X ′ ) ∈ σ C. ⃗ be topological closed-hereditary and closed-additive ∪∞(we may ′suppose ∪∞ C = ′ SU (M, X)∩SU (M, X )). Then we can write Z = n=1 Zn , Z = n=1 Zn′ , ′ are Z-sets in M such that where for every n ∈ N Zn ⊂ Zn+1 , Zn′ ⊂ Zn+1 ′ ′ ′ ⃗ (Zn , Zn ∩ X), (Zn , Zn ∩ X ) ∈ C. In order to ﬁnd a homeomorphism f : (M, X) → (M, X ′ ) close to id, we shall construct inductively sequences of homeomorphisms {fn : M → M } and {gn : M → M } satisfying for every n the following conditions: −1 −1 −1 (1n ) (fn , gn−1 ) ≺ Un+1 , (fn , gn−1 ) ≺ gn−1 Un+1 ;

1.7. STRONGLY UNIVERSAL AND ABSORBING PAIRS

(2n ) (3n ) (4n ) (5n ) (6n )

47

−1 ′ ′ fn |Zn−1 ∪ gn−1 (Zn−1 ) = gn−1 |Zn−1 ∪ gn−1 (Zn−1 ); −1 ′ (fn |Zn ) (X ) = Zn ∩ X; (gn , fn−1 ) ≺ Un+1 , (gn , fn−1 ) ≺ fn−1 Un+1 ; ′ ′ gn |Zn−1 ∪ fn (Zn ) = fn−1 |Zn−1 ∪ fn (Zn ); ′ −1 ′ ′ (gn |Zn ) (X) = Zn ∩ X ;

Letting f0 = g0 = id we proceed inductively. Assume that homeomorphisms fi , gi (satisfying (1i )—(6i ) for i = 1, . . . , n − 1) have been constructed. It follows from Z-Set Unknotting Theorem for Q- or s-manifolds that there is a cover W ∈ cov(M ) such that every Z-embedding h : Zn ∪ −1 ′ ′ gn−1 (Zn−1 ) → M W-close to gn−1 |Zn ∪ gn−1 (Zn−1 ) can be extended to −1 ¯ ¯ ¯ g −1 ) ≺ a homeomorphism h of M such that (h, gn−1 ) ≺ Un+1 and (h, n−1 −1 gn−1 Un+1 . ′ ′ Let K = Zn ∪ gn−1 (Zn−1 ) and B = Zn−1 ∪ gn−1 (Zn−1 ). It follows ′ ′ from (6n−1 ) that K ∩ X = (Zn ∩ X) ∪ gn−1 (Zn−1 ∩ X ). Since the class ⃗ It follows from (2i ), C⃗ is topological and closed-additive, (K, K ∩ X) ∈ C. −1 (3i ), (5i ), (6i ), i < n, that gn−1 |B : B → M is a Z-embedding with −1 ⃗ (gn−1 |B)−1 (X) = B ∩ X ′ . By the strong C-universality of (M, X ′ ) there −1 exists a Z-embedding h : K → M such that h|B = gn−1 |B, h−1 (X ′ ) = −1 K ∩ X, and (h, gn−1 |K) ≺ W. By the choice of W, the Z-embedding h can be extended to a homeomorphism fn : M → M such that satisﬁes the condition (1n ). The conditions (2n ), (3n ) are satisﬁed as it follows from the construction of fn . Using the same reasoning, construct a homeomorphism gn satisfying the conditions (4n )—(6n ). The inductive step is over. Remark that by (1n ), (4n ), n ∈ N, we have (fn , fn−1 ) ≺ Un and (gn , gn−1 ) ≺ Un . Thus the limit maps f = limn→∞ fn : M → M and g = limn→∞ gn : M → M are well-deﬁned, continuous, and such that (f, id)∪≺ U0 , (g, id) ≺ U0 . One∪can deduce from (2n ), (5n ), n ∈ N, that ∞ ∞ f ◦ g| n=1 Zn′ = id and g ◦ f | n=1 Zn = id. Without loss of generality, ⃗ and consequenly X ̸= ∅. In this case, the pair ({∗}, {∗}) belongs to C, ′ both X and X must be dense in M (because the pairs (M, X), (M, X ′ ) ∪∞ ∪∞ are strongly ({∗}, {∗})-universal). Then n=1 Zn ⊃ X and n=1 Zn′ ⊃ X ′ are dense in M , and thus f ◦ g = id, g ◦ f = id. By (3n ), (6n ), n ∈ N, f −1 (X ′ ) = X, i.e., f is a homeomorphism with f (X) = X ′ . Sometimes we will need a little bit more general version of 1.7.6. 1.7.7. Theorem. Let C⃗ be a class of pairs, M either a Q-manifold or an ⃗ s-manifold, (M, X), (M, X ′ ) two C-absorbing pairs, and B a closed subset ′ in M such that B ∩ X = B ∩ X . For every cover U ∈ cov(M ) there is a homeomorphism h : M → M such that (h, id) ≺ U , h|B = id|B, and h(X) = X ′ . Proof. Replacing if necessary C⃗ by SU (M, X) ∩ SU (M, X ′ ) we may as-

48

BASIC THEORY

sume that C⃗ is a topological closed-hereditary closed-additive class. Under this assumption, one can easily show that the pairs (M \B, X\ B) and ⃗ (M \B, X ′ \B) are C-absorbing. Applying Theorem 1.7.6 to these pairs, we can ﬁnd a homeomorphism h′ : M \B → M \B such that h(X\B) = X ′ \B and (h′ , id) ≺ U ′ , where U ′ ∈ cov(M \B) is any cover with U ′ ≺ U , U ′ ≺ {B(x, d(x, B)/2) | x ∈ M \B} (here d is a metric on M ). Letting h|B = id, h|M \B = h′ , we extend h′ to a homeomorphism h : M → M satisfying our requirements. We deﬁne a pair (M, X) to be absorbing, provided (M, X) is F0 (M, X)absorbing. ⃗ 1.7.8. Proposition. Let either M = Q or M = s. Every C-absorbing pair (M, X) is absorbing. Proof. To prove the proposition, we have to verify the strong F0 (M, X)-universality of (M, X). For this, ﬁx a pair (F, F ∩ X), where F is a closed subset in M , a cover U ∈ cov(M ), a closed subset B ⊂ F , and a map f : F → M such that the restriction f |B : B → M is a Z-embedding with (f |B)−1 (X) = B ∩ X. Without loss of generality, C⃗ is a topological closed-hereditary class. By ⃗ the deﬁnition of C-absorbing pair, there is a Zσ -set Z ⊂ M such that ⃗ (Z, X) ∈ σ C. Let V ∈ cov(M ) be a cover with St V ≺ U. By 1.1.21 or 1.1.26, there is a Z-embedding f ′ : F → M such that (f ′ , f ) ≺ V, f ′ |B = f |B and f ′ (F \B) ∩ Z = ∅. By Proposition 1.7.5, the pair (M, X ∪ f ′ (F ∩ X)) is ⃗ strongly C-universal. It is easily seen that Z ∪ f ′ (F ∩ Z) is a Zσ -set in M ′ ⃗ Then by Theorem 1.7.7, there such that (Z ∪f (F ∩Z), X ∪f ′ (F ∩X)) ∈ σ C. is a homeomorphism h : M → M such that (h, id) ≺ V, h|f (B) = id|f (B) and h−1 (X) = X ∪ f ′ (F ∩ X). Letting f¯ = h ◦ f ′ : F → M we see that (f¯, f ) ≺ St V ≺ U, f¯|B = f |B and f¯−1 (X) = F ∩ X, i.e. f¯ is the required Z-embedding. Now let us prove a theorem revealing connections between strongly universal pairs and strongly universal spaces. 1.7.9. Theorem. Let (K, C) be a pair, M an ANR, and X a homotopy dense subspace in M such that X has SDAP. If the pair (M, X) is strongly (K, C)-universal then the space X is strongly C-universal. Proof. Suppose (M, X) is a strongly (K, C)-universal pair. According to 1.5.7, to show that X is strongly C-universal, it suﬃces for given open ◦

◦

◦

◦

◦

sets C ⊂ C, X ⊂ X, cover U ∈ cov(X), and map f : C → X to ﬁnd a ◦ ◦ Z-embedding f¯ : C → X, U-close to f . ◦

Let V ∈ cov(X) be a cover with St 2 V = St (St V) ≺ U . By Proposition ◦

◦

◦

◦

1.1.12, there exists a map f ′ : K → X of an open neighborhood K of C in

1.7. STRONGLY UNIVERSAL AND ABSORBING PAIRS ◦

◦

49

◦

◦

K such that (f ′ |C, f ) ≺ V. The set K can be chosen so that K ∩ X = C. ◦ ◦ ∪∞ Write K = n=1 Kn , where each Kn ⊂ Int Kn+1 ⊂ K is a closed subset in ◦

◦

K. By Propositions 1.3.3 and 1.3.4, there is a map f0 : K → X such that (f0 , f ′ ) ≺ V and the collection {f0 (Kn+1 \ Int Kn−1 )}n∈N is locally ﬁnite in ◦

◦

X (here we put K0 = ∅). Using Ex.3 to §1.1.A, ﬁnd a cover W ∈ cov(X) such that St W ≺ V and the collection {St (f0 (Kn+1 \ Int Kn−1 ), St W)}n∈N ◦

is locally ﬁnite in X. ˜ in M with W ˜ ∩ X = W and For every W ∈ W ﬁnd an open sets W ◦ ∪ ˜ in M and the cover W ˜ = {W ˜ |W ∈ consider the open set M = W ∈W W ◦

◦

◦

W} of M . Notice that X = X ∩ M . Similarly as in the proof of Lemma 1.5.9, construct inductively a sequence ◦

◦

of maps fn : K → M , n ∈ N, satisfying the following conditions ◦

˜ fn = fn−1 on Kn−1 ∪ (K\ Int Kn+1 ), and (fn , fn−1 ) ≺ W, ◦

fn |Kn : Kn → M is a Z-embedding with (fn |Kn )−1 (X) = Kn ∩ C. ◦

◦

◦

◦

◦

Let ﬁnally f˜ = limn→∞ fn : K → M and f¯ = f˜|C : C → X. We claim that the map f¯ is a Z-embedding with (f¯, f ) ≺ U . Indeed, noticing that ˜ we obtain that for any n ∈ N (f˜, f0 ) ≺ St W, ◦

f¯(C ∩ (Kn+1 \ Int Kn−1 )) ⊂ St (f0 (Kn+1 \ Int Kn−1 ), St W). ◦

◦

Hence, the collection {f¯(C ∩ (Kn+1 \ Int Kn−1 ))}n∈N is locally ﬁnite in X. By the construction, for every n ∈ N the map f˜|Kn = fn |Kn is a Z◦

◦

◦

embedding. By 1.2.3, the set X = X ∩ M is homotopy dense in M . Thus ◦ ◦ ◦ f¯(Kn ∩ C) = f˜(Kn ) ∩ X is a Z-set in X (see Ex.16 to §1.4). Consequently, ◦

for every n ∈ N the restriction f¯|(Kn+1 \ Int Kn−1 ) ∩ C is a Z-embedding. By Ex.6 to §1.4, the map f¯ is a Z-embedding. The second condition, namely (f¯, f ) ≺ U , easily follows from (f, f0 ) ≺ St V, (f¯, f0 ) ≺ St W, St W ≺ St V and St 2 V ≺ U. Finally, let us prove the following useful result. 1.7.10. Proposition. Let M be an ANR, X ⊂ M , Y a homotopy dense subset of M , K a compact space, and C ⊂ K. If the pair (Y, Y ∩ X) is strongly (K, C)-universal then so is the pair (M, X). Proof. Fix a closed subset B ⊂ K, a cover U ∈ cov(M ), and a map f : K → M whose restriction on B is a Z-embedding with (f |B)−1 (X) = B ∩C. Pick

50

BASIC THEORY

a cover U ′ ∈ cov(M ) such that St U ′ ≺ U , and a cover U ′′ ∈ cov(M \f (B)) such that U ′′ ≺ U ′ and U ′′ ≺ {B(x, d(x, f (B))/2) | x ∈ M \f (B)}. Using the fact that f (B) is a Z-set and Y is homotopy dense in M , construct a map f ′ : K → M such that f ′ |B = f |B, (f ′ , f ) ≺ U ′ , and f ′ (K\B) ⊂ Y \f (B). Using strong F0 (K, C)-universality of (Y, Y ∩ X), and 1.7.4, construct a Z-embedding g : K\B → Y \f (B) such that (g, f ′ |K\f (B)) ≺ U ′′ and g −1 (Y ∩ X) = C\B. Then the map f¯ : K → M deﬁned by f¯|B = f |B and f¯|K\B = g is a Z-embedding (K is compact!) such that (f¯, f ) ≺ U , f¯|B = f |B, and f¯−1 (X) = C.

Exercises to §1.7. 1. (Deleting Theorem). Let (X, Y ) be a strongly (K, C)-universal pair, where X is an ANR such that every Z-set in X is strong. Show that for every subset A ⊂ X whose ¯ is a Z-set in X the pair (X, Y \A) is strongly (K, C)-universal. closure A 2. Show that a pair (M, X) is strongly (K, C)-universal iﬀ the pair (M, M \X) is strongly (K, K\C)-universal. 3. Suppose X is an ANR and Y ⊂ X. Prove that if the pair (X, Y ) is strongly (I n , I n )universal for every n ∈ N then the set Y is homotopy dense in X. ⃗ 4. Let (M, X) be a C-absorbing pair, where C⃗ is a class of pairs, and M is a Q-manifold. Show that SU (M, X) = {(K, C) ∈ F0 (M, X) | K is compact}.

5. Suppose X is an ANR such that every Z-set ∪ in X is strong, and X1 ⊂ X2 ⊂ . . . ∞ ⃗ where C⃗ = is a K-skeleton in X. Show that the pair (X, n=1 Xn ) is C-absorbing, {(K, K) | K ∈ K}. For classes of spaces K, C let (K, C) = {(K, C) | C ∋ C ⊂ K ∈ K}. 6. Show that the pairs (Q, σ), (s, σ), and (l2 , lf2 ) are (M0 , M0 (ω))-absorbing and the pairs (Q, Σ), (s, Σ) are (M0 , M0 )-absorbing. ∼ 7. Using Theorem 1.7.10, show that for every Zσ -subset A ⊂ Q, if Σ ⊂ A then (Q, A) = (Q, Σ), and if additionally A ⊂ s then (s, A) ∼ = (s, Σ). 8. Show that for every Zσ -subset Z ⊂ Q there is a homeomorphism h : Q → Q with h(Z) ⊂ s. 9. Suppose M ∈ ANR is a co-Zσ -space satisfying LCAP (e.g., M is a Q-manifold or an s-manifold) and X a homotopy dense subset in M . Assuming that the pair (M, X) is absorbing show that the space X is absorbing and its complement M \X is a coabsorbing space. 10. Find a subset X in the Hilbert cube Q such that the pair (Q, X) is absorbing but neither X is absorbing nor Q\X is coabsorbing. Hint: Let X be any countable dense set in Q and use Ex.5 to show that the pair (Q, X) is absorbing. 11. Let X be an ANR with LCAP, Y ⊂ X, and C⃗ a class of pairs. Suppose X contains a tower X1 ⊂ X2 ⊂ · · · ⊂ Xn ⊂ . . . of Z-sets in X such that a) each Xn is an ANR; ⃗ b) the pair (Xn , Xn ∩ Y ) is strongly C-universal for every n; c) given a cover U ∈ cov(X), n ∈ N, and a map f : K → X of a ﬁnite-dimensional compactum, there exist an m ∈ N and a map f¯ : K → Xm such that (f¯, f ) ≺ U and f¯|f −1 (Xn ) = f |f −1 (Xn ).

NOTES AND COMMENTS TO CHAPTER I

51

⃗ Show that the pair (X, Y ) is strongly C-universal. ⃗ 12. Let either M = Q or M = s, and let (M, Ω) be an C-absorbing pair, where C⃗ is a class of pairs. A pair (X, Y ) is said to be an (M, Ω)-manifold if every point x ∈ X has a neighborhood U ⊂ X such that (U, U ∩ Y ) ∼ = (V, V ∩ Ω) for some open subset V ⊂ M . Show that a pair (X, Y ) is an (M, Ω)-manifold iﬀ X is an M -manifold and the pair ⃗ (X, Y ) is C-absorbing. Let (Γ, ≤) be a partially ordered set. A collection (Xγ )γ∈Γ of subsets of a space X is deﬁned to be a Γ-system in X if Xγ ⊂ Xγ ′ for every γ ≤ γ ′ . A Γ-system, by deﬁnition, is a pair (X, Xγ )γ∈Γ consisting of a space X and a Γ-system in X. Partial cases of Γ-systems are spaces (if Γ = ∅) and pairs (if |Γ| = 1). 13. Generalize results of sections 1.5–1.7 to the case of Γ-systems. Hint: see J.Baars, H.Gladdines, J. van Mill [1993]

Notes and Comments to Chapter I The reader is referred to K.Borsuk [1967], S.T.Hu [1965], C.Bessaga, A.Pelczy´ nski [1975], T.Chapman [1976], V.Fedorchuk, A.Chigogidze [1992], and J. van Mill [1989] for the historical comments concerning the theory of retracts and the theory of Q- and s-manifolds. Here we only note that Characterization Theorems 1.1.14 and 1.1.23 are due to H.Toru´ nczyk [1980], [1981]. The notion of homotopy dense/negligible set is a derivative of that of locally homotopy negligible set; see the paper of Toru´ nczyk [1978], where one can also ﬁnd the material related to Ex.12–14 to §1.2. For the origins of the Strong Discrete Approximation Property see H. Toru´ nczyk [1981] and M.Bestvina et al [1986]. Proposition 1.3.1 is wellknown to specialists; see, e.g. D.Curtis [1985]. Theorem 1.3.2 is due to T.Banakh [1998a], Theorem 1.3.4 is due to T.Banakh and R.Cauty [2000]. See M.Bestvina et al [1986] for Ex.11 to §1.3. The Locally Compact Approximation Property (Ex.12 to §1.3) is introduced and investigated by T.Banakh [1998b]. The notion of Z-set is due to R.Anderson [1967]. Example 1.4.2 is taken from M.Bestvina et al [1986]. Almost all the results of §1.4 have their origins in the papers of D.Henderson [1975] and M.Bestvina, J.Mogilski [1986]. The Strong Universality is introduced by M.Bestvina and J.Mogilski [1986]. The results of §1.5 can be found in M.Bestvina, J.Mogilski [1986] and T.Banakh, R.Cauty [2000]. The notion of C-absorbing space is a generalization of a cap set of R.Anderson [197?]; it is introduced by M. Bestvina and J.Mogilski [1986] for subsets of s-manifolds. Theorem 1.3.2 allows us to deﬁne the equivalent absolute notion (Deﬁnition 1.6.1; see also T.Dobrowolski, J.Mogilski [1990a]) and to simplify essentially the original proof of Theorem 1.6.3 given

52

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in M.Bestina, J.Mogilski [1986]. The notion of C-coabsorbing set has appeared in T.Banakh, R.Cauty [2000]; Theorem 1.6.4 is due to T.Banakh. The notion of strongly universal pair as well as Uniqueness Theorem 1.7.6 is due to R.Cauty [1991a]. Theorem 1.7.9 is due to T.Banakh, R.Cauty [2000] and Theorem 1.7.10 to J.Baars, H.Gladdiness, J. van Mill [1993].

53

Chapter II Constructions of absorbing spaces

In this chapter we consider the problem of existence of C-absorbing spaces and pairs for diﬀerent classes C of spaces. Mainly, these classes are deﬁned by means of conditions taken from the descriptive theory, and we give brief surveys of them in Sections 2.1, 2.2. The main results are given in Sections 2.4—2.6. §2.1. Preliminaries I. Descriptive set theory. For a space X we deﬁne the additive Aα (X) and multiplicative Mα (X) classes of Borel subsets of X, corresponding to a countable ordinal α, as follows. Let A0 (X) (resp. M0 (X)) denote the family of all open (resp. closed) subsets of X; for a countable ordinal α > 0 let ∪

Aα (X) = {∪∞ i=1 Xi | Xi ∈

Mβ (X)},

β N . Assume the converse. Since Ω ∈ Cσ , Baire Theorem allows us to ﬁnd an open set U ⊂ C L such that U emN q beds in C L = ∏∞ into LC × I for some q ∈ N. The set U being ∏∞ open j L ( i=1 Ci ) contains a topological copy of the product ( j=r C ) for some ∏∞ ∏L ∏∞ r ∈ N. Write ( j=r Cj )L = m=1 j=r Cm,j , where Cm,j = Cj , and ∏N ∏∞ C N = n=1 i=1 Cn,i , where Cn,i = Ci , and let φ denote the embedding ∏N ∏∞ ∏∞ of ( j=r Cj )L into C N × I q = ( n=1 i=1 Cn,i ) × I q . Denote by π1 : C N × I q → C N , π2 : C N × I q → I q , pn,i :

N ∏ ∞ ∏

Cn,i → Cn,i = Ci

n=1 i=1

and ρm,j :

∏L m=1

∏∞ j=r

Cm,j → Cm,j = Cj the corresponding projections.

Claim. For every (n, i) ∈ {1, . . . , N } × N the map pn,i ◦ π1 ◦ φ is either constant or coincides with the projection ρm,i for some m ∈ {1, . . . , L}. Assume that pn,i ◦ π1 ◦ φ is not constant. Then we can ﬁnd two points ∏L ∏∞ y, y ′ ∈ m=1 j=r Cm,j which diﬀer by one coordinate only (to say by (m, j)-coordinate) such that pn,i ◦ π1 ◦ φ(y) ̸= pn,i ◦ π1 ◦ φ(y ′ ). Write ∏L ∏∞ m=1 j=r Cm,j = Cm,j × Z. For every z ∈ Z deﬁne the map ψz : Cm,j = Cj → Ci = Cn,i by ψz (c) = pn,i ◦ π1 ◦ φ(c, z). By properties of Cook’s continuum L and by connectedness of Z either ψz is constant for all z ∈ Z or ψz = id|Cj for all z ∈ Z. Notice that the ﬁrst case is impossible because writing y = (c, z), y ′ = (c′ , z) we have ψz (c) = pn,i ◦ π1 ◦ φ(y) ̸= pn,i ◦ π1 ◦ φ(y ′ ) = ψz (c′ ). Therefore pn,i ◦ π1 ◦ φ = ρm,j . Notice also that in this case necessarily j = i. Denote by I0 the set of all pairs (m, j) for which there is an n ∈ {1, . . . , N } with pn,j ◦ π1 ◦ φ = ρm,j . Since N < L, the set I1 = ({r, r + 1, . . . } ×

2.4. ABSORBING SPACES FOR I-STABLE CLASSES

85

∏ {1, . . . L})\I0 is inﬁnite. Fixing any point z ∈ (m,j)∈I0 Cm,j we obtain ∏ that the restriction of π1 ◦ φ onto B = (m,j)∈I1 Cm,j × {z} is constant. Thus π2 ◦ φ|B : B → I q is injective. This is impossible because B is inﬁnitedimensional. As applications of Theorem 2.4.2, we will prove the following results. 2.4.7. Theorem. For each countable ordinal α there exist a countable ordinal ξ ≥ α and an M0 (ind, ξ)-absorbing space. Proof. By Theorem 2.2.53, there exist a countable ordinal ξ ≥ α and a compactum Kξ ∈ σ-M0 (ind, ξ) such that Kξ contains a topological copy of ¯ every X ∈ M0 (ind, ξ) and k(ξ) = ξ. The latter condition implies that the class M0 (ind, ξ) is [0, 1]-stable. Since F0 (Kξ ) ⊃ M0 (ind, ξ), we can apply Theorem 2.4.2. 2.4.8. Theorem. Let C ∈ B. There exists a C-absorbing set Ω(C). Proof. This follows from Theorem 2.4.2 and 2.1.3.

The following notation is used for Ω(C): Λα if C = Aα , Ωα if C = Mα , Λβα if C = Aβα , Ωβα if C ∈ Mβα , Πi if C = Pi . Using Ex.7 to §1.3, for every such Ω(C) ﬁx a homotopy dense embedding Ω(C) ⊂ Q (in §3.2 we will show that for majority of classes C all such embedding are topologically equivalent). 2.4.9. Theorem. Let C ∈ B. There exists a C(s.c.d.)-absorbing space.

2.4.10. Theorem. Let C ∈ B\{A1 , M1 , M2 }. There exists a C(c.d.)absorbing space. 2.4.11. Theorem. Let C ∈ B\{A1 }. For every countable abelian group G there exists a C(dimG , ω)-absorbing space. Exercises to §2.4. 1. Suppose that for i = 1, 2 Ci is an I-stable M1 -additive closed-hereditary class with Ci = σCi , and Xi ∈ Ci is a Ci -coabsorbing space. Show that the product X1 × X2 is a coabsorbing space. 2. Let C1 be an I-stable M1 -additive closed-hereditary class and C2 be an I-stable closedhereditary class with C2 = σC2 . Assuming that X1 is a C1 -absorbing space and X2 ∈ C2 a C2 -coabsorbing, prove that the product X1 × X2 is an absorbing space. ∼ Σ, Λ1 (s.c.d) = ∼ σ, Ω1 = ∼s×σ = ∼ s × Σ and Ω2 = ∼ Σω = ∼ σω . 3. Show that Λ1 = 4. Suppose X is a σ-compact AR with SDAP. Prove that X × Q ∼ = Λ1 . 2 5. Suppose X ∈ AR is a σZ-space of the class M . Show that X × s ∼ = Ω1 . 1

86

CONSTRUCTIONS OF ABSORBING SPACES

§2.5. Weak inverse limits and absorbing spaces. In this section we describe a construction of absorbing spaces by means of the operation of weak inverse limit. An inverse sequence S = {Xi , pij } is called coretractible if for each i there exists an embedding si : Xi → Xi+1 such that pi+1 si = 1Xi . We i S. By the weak inverse limit denote this by S = {Xi , pij ; sj }. Let X = lim ←− of S we mean the set w-lim S = {(xi )∞ ∈ lim S | there exists j ∈ N i=1 ←− such that xi+1 = si (xj ) for each i ≥ j} endowed with topology inherited from X. For each j we have the natural embedding ιj : Xj →w-lim S j deﬁned by the condition: ιj (x) = (yi )∞ i=1 where yi = p∪ i (x) if i ≤ j and ∞ yi = (si−1 ◦ · · · ◦ sj )(x) otherwise. Obviously, w-lim S = i=1 ιj (Xj ). Denoting by pj : X → Xj the limit projections of S we obtain pj ιj = 1Xj . 2.5.1. Proposition. Let C be a class of spaces and S = {Xi , pji ; si } a coretractible inverse sequence such that all bonding maps pi+1 are C-soft. i Then the restriction of the limit projection pi onto X ′ =w-lim S is C-soft. Proof. Consider a commutative diagram φ

A −−−−→ w- lim S pj incly y Z −−−−→ ψ

Xj

where A is a closed subset of a space Z ∈ C. Let Z\A = ∪{Bi | i ≥ j} where Bj ⊂ Bj+1 ⊂ . . . is a sequence of closed subsets in Z. We have φ = (φi )∞ i=1 where φi : A → Xi are the coordinate maps. Deﬁne maps Φi : Z → Xi , i ≥ j by induction. Let Φj = ψ and assume that for each i, j ≤ i < k maps Φi : Z → Xi are deﬁned. By C-softness of pkk−1 there exists a map Φk such that pkk−1 Φk = Φk−1 , Φk |A = φk and ′ Φk |Bk = ιk−1 Φk−1 |Bk . Clearly, the map Φ = (Φi )∞ i=1 : Z → X is welldeﬁned and Φ|A = φ, pj Φ = ψ. Assume that C is a topological, closed-hereditary class which contains the class of ﬁnite-dimensional compacta and for which there exists a map f : X → Ω with the following properties: (i) Ω ∈ AR contains a copy of the Hilbert cube and is σC-universal; there is a point ∗ ∈ Ω such that {∗} is a Z-set in Ω; (ii) f is σC-soft; (iii) X ∈ σC and X ∈ AR. We construct (noncanonically) a coretractible inverse sequence S = {Xi , pi+1 ; si } by the following manner. i

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Let X1 = X and suppose that the spaces Xi and maps pii−1 : Xi → Xi−1 are already deﬁned for every i < j. Consider the space Xj−1 × Ω as a subspace in Ω (here we use the fact that Ω contains a copy of the Hilbert cube) and let Xj = f −1 (Xj−1 × Ω), pjj−1 = pr1 (f |Xj ). Assume additionally that (iv) Xj ∈ σC for every j ∈ N. By σC-softness of f there exists a map si : Xi → Xi+1 such that f (si (x)) = (x, ∗) for every x ∈ Xi . ; si }. We will identify the spaces Xi with the Let Ωf =w-lim{Xi , pi+1 i corresponding subspaces of Ωf along the natural embeddings ιj given by the formula ιj (x) = (yi )∞ i=1 ∈ Ωf where i+1 j−1 pi ◦ · · · ◦ pj (x) if j > i, yi = x if j = i, si−1 ◦ · · · ◦ sj (x) if j < i, Let pj : Ωf → Xj denote the restriction of the limit projection lim ←− S → Xj onto Ωf . We endow Ωf with the metric d: d((xi ), (yi )) =

∞ ∑ di (xi , yi ) i=1

2i

where di is a metric on Xi restricted by 1. 2.5.2. Proposition. For each j ∈ N the set Xj is a Z-set in Ωf . Proof. Without restricting generality we assume that j = 1. Let g : I k → Ωf be a map. Consider a map (f |X2 )p2 g : I k → X1 × Ω. There exists a sequence of maps hl : I k → X1 × (Ω\{∗}), l ∈ N, uniformly convergent to (f |X2 )p2 g. Let S = {0} ∪ {1/l | l ∈ N} be the convergent sequence. Deﬁne a map ˆ h : I k × S → X1 × Ω by the formula h(x, 0) = (f |X2 )p2 g(x), h(x, 1/l) = hl (x), x ∈ I k , l ∈ N. It follows from σC-softness of the map f that there exists a map r : I k ×S → X2 such that (f |X2 )r = h, r(x, 0) = p2 g(x), x ∈ I k . Since the map p2 : Ωf → X2 is σC-soft, there exists a map r′ : I k × S → Ωf such that p2 r′ = r, r′ (x, 0) = g(x), x ∈ I k . By compactness of I k there exists l ∈ N such that d(r′ (x, l), g(x)) < ε for each x ∈ I k . Deﬁne a map g ′ : I k → Ωf by the formula g ′ (x) = r′ (x, l), x ∈ I k . Then, obviously, g ′ (I k ) ∩ X1 = ∅.

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2.5.3. Proposition. Every open subset U of Ωf satisﬁes the property: every map g : C → U where C ∈ σC can be approximated by Z-embeddings. Proof. Let U ∈ cov(U ). There exists a function α : U → (0, 1) such that α(x) < 12 d(x, Ωf \ U ) and α-close maps are U-close (if U = Ωf , then the latter condition is superﬂuous). Assume that C ∈ σC and a map g : C → U is given. We write g = (gi )∞ i=1 where gi : C → Y are the coordinate maps. Deﬁne maps gi′ : C → Yi by induction. Put g1′ = g1 and assume that for each j ≤ n maps gj′ are already deﬁned. Since Ω ∈ AR, there exists a map h : Ω × Ω × Ω × [0, +∞) → Ω with the following properties: h(a, b, c, t) =

a if t ≤ 1 b if 2 ≤ t ≤ 8 c if t ≥ 16.

Denote by φ : C → Ω a closed embedding such that ∗ ∈ / φ(C). Deﬁne the map gˆn+1 : C → Yn × Ω by the formula gˆn+1 (c) = (gn′ (c), h(π2 f gn+1 (c), ϕ(c), ∗, 2n αg(c))). (π2 denotes the projection map onto the second factor). Note that for each c ∈ (αg)−1 ((0, 2−n ]) we have f gn+1 (c) = gˆn+1 (c) and hence, by C-softness ′ : C → Yn+1 satisfying the properties: of f , there exists a map gn+1 ′ 1)n f gn+1 = gˆn+1 ; ′ |(αg)−1 ((0, 2−n ]) = gn+1 |(αg)−1 ((0, 2−n ]); 2)n gn+1 ′ 3)n gn+1 |(αg)−1 ([2−n+4 , 1]) = sn gn′ |(αg)−1 ([2−n+4 , 1]). Put g ′ = (gn )∞ n=1 . It follows from the property 3)n that for each c ∈ C such that αg(c) ≥ 2−n+4 we have g ′ (c) ∈ X. Hence g ′ (C) ⊂ Ωf . The conditions 2)n imply that the maps g and g ′ are α-close, i.e., g ′ (C) ⊂ U. Show that g ′ is a closed embedding. Indeed, assume the contrary. Then there exists a closed discrete subspace {ci | i ∈ N} of C such that its image under the map g ′ is not closed in Ωf . Without loss of generality we can assume that there exists y = limi→∞ g ′ (ci ), y ∈ Ωf . Passing, if necessarily, to a subsequence we can also assume that there exists a = limi→∞ α(g(ci )). Case 1. a = 0. Then limi→∞ α(g ′ (ci )) = 0 and α(y) = 0. Contradiction. Case 2. There exists j ∈ N such that a ∈ (2−j−1 , 2−j+1 ). Note that the ′ map gj+1 |(αg)−1 ([2−j−1 , 2−j+1 ]) is a closed embedding, and therefore so is

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the map g ′ |(αg)−1 ([2−j−1 , 2−j+1 ]). Then we have (g ′ )−1 (y) = limi→∞ ci which gives a contradiction. Since each Xi ∩ U is a Z-set in U (see Corollary), the set g ′ (C) is a local Z-set in U . By Proposition 1.4.8, g ′ (C) is a Z-set in U . Now we are able to prove the main result of this section. 2.5.4. Theorem. Let a class C and a map f : X → Ω be as above. Then Ωf is a C-absorbing space. Proof. First prove that Ωf ∈ AR. Regard Ωf as a closed subset of an AR-space Y such that Y \ Ωf is a countable simplicial complex. By the assumption on C, ﬁnite-dimensional simplexes are in C and therefore we have Y ∈ σC. Since X1 ∈ AR, there exists an extension p¯1 : Y → X1 of the limit projection p1 . By σC-softness of p1 , there exists a map r : Y → Ωf such that r|Ωf = 1Ωf . Thus, the space Ωf , being a retract of the AR-space ⊔∞ Y , is an AR-space as well. Since i=1 I i ∈ σC, it follows from Proposition 2.5.3 that the space Ωf satisﬁes SDAP. Thus, by Proposition 2.5.3, every Z-set in Ωf is a strong Z-set. By 1.5.7 and Proposition 2.5.3, the set Ωf is σC-absorbing. 2.5.5. Theorem. Let C ⊂ B ∈ B\{A1 } be a class of spaces which contains a class of ﬁnite-dimensional compacta and satisﬁes the conditions: (i) C is B-hereditary; (iii) C is 2ω -stable; (iv) there exists a space Y ∈ σC containing a copy of each space X ∈ C. Then there exists a C-absorbing space Ω(C). Proof. It follows from Theorem 2.3.10 that there exists a space S ∈ σC and C-invertible map g : S → Q. Since C contains a class of ﬁnite-dimensional compacta, we can assume that S ∈ AR. Denote by Ω the B-absorbing space from Theorem 2.4.8. It follows from Theorem 2.3.22 that there exist a σC-soft map f˜ : S → Q. Consider the space Ω as subspace of Q and let X = f˜−1 (Ω) and f = f˜|X. It is easy to check that X ∈ AR. Since the class C is B-hereditary, we have that X ∈ σC. Now we can use the construction Ωf to obtain a C-absorbing space Ω(C). 2.5.6. Corollary. Let C ∈ B\{A1 } and F one of dimension functions ind, zwS , zwH . For each countable ordinal α there exists a C(F, α)-absorbing space. These absorbing spaces are not σ-compact. The following result is an application of the above technique to the σ-compact case. 2.5.7. Theorem. There exists a σN -absorbing space.

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CONSTRUCTIONS OF ABSORBING SPACES

There exists a modiﬁcation of the above construction for pairs of spaces. For the sake of simplicity, we consider only the σ-compact case. Further, C is a subclass of A1 . The following lemma is a slight strengthening of Proposition 2.3.14. 2.5.8. Lemma. Suppose that there exists a C-soft map f : X → Σ where X ∈ C. For each closed subset A ⊂ Σ, A ∈ C there exists an C-soft map g : Y → Σ satisfying the following properties: (i) g|g −1 (A) : g −1 (A) → A is a homeomorphism; (ii) for every map h : C → Σ\A where C ∈ C there exists an embedding h′ : Y → X such that gh′ = h; (iii) Y ∈ σC. Proof. Consider the composition X −−−−→ Σ ∼ = Σ × Σ −−−−→ Σ. f

pr1

and apply to it the construction of the proof of Proposition 2.3.14. Theorem 2.5.9. Let C ⊂ A1 be a class for which there exists a C-soft map l : Z → Σ where Z ∈ C. Then there exists an (M0 , C)-absorbing pair. Proof. We consider a commutative diagram Q ←−−−− Q × Q ←−−−− Q × Q × Q ←−−−− . . . p21 p32 x x x Σ ←−−−− Σ × Σ ←−−−− Σ × Σ × Σ ←−−−− . . . p21 p32 x x x q1 q2 q3 Z1 −−−−→ s1

Z2

−−−−→ s2

Z3

−−−−→ . . .

satisfying the properties: (1) qi : Zi → Σi = Σ × · · · × Σ (i factors) is a C-soft map where Zi ∈ σC; (2) qi |qi−1 (Zi−1 ) : qi−1 (Zi−1 ) → Zi−1 is a homeomorphism and qi+1 si = id; (3) for every map g : Y → Σ\Zi−1 where Y ∈ σC there exists an embedding h : Y → Zi such that gi h = g; (4) pi+1 : Σi+1 → Σi is the projection; i (5) Zi is a subset of Σi × K where K ⊂ Σ is a Z-set in Q and pi+1 |Zi = qi . i The limit space of the upper row inverse sequence is naturally identiﬁed with Qω . ω Let Xi = {(xj )∞ j=1 ∈ Q | (x1 , . . . , xj+1 ) = sj (x1 , . . . , xj ) for each j ≥ i} ∞ and X = ∪i=1 Xi . Obviously, Xi is closed in X and Xi is homeomorphic to Zi for each i. Hence, X ∈ σC.

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Denote by pi : X → Xi the natural projection. We are going to prove that (Qω , X) is an (M0 , C)-absorbing pair. Let (C, D) be a pair of spaces where C is compact, D ∈ C, and f = (fi )∞ i=1 : C → Qω a map such that the restriction f = (fi )∞ i=1 |A onto a closed subset A of C is a Z-embedding and (f |A)−1 (X) = A ∩ D. Note that the set Qω \Σω is locally homotopy negligible in Qω , i.e., there exists a homotopy (ht ) : Qω → Qω such that h0 = id and ht (Qω ) ⊂ Σω for every t ∈ (0, 1]. Taking a function φ : Qω → [0, 1] with φ−1 (0) = f (A) we obtain a homotopy (gs ) : Qω → Qω deﬁned as follows: gs (x, t) = hsφ(x) (x). Then g0 = id and gs (Qω ) ⊂ f (A) ∪ Σω . Since f (A) is a Z-set in Qω , we may assume, without restricting generality, that f (C\A) ⊂ Σω \f (A). Let ε > 0 and α : C → [0, +∞) be a function deﬁned by the formula: α(c) = min{d(f (c), f (A)), ε}. We deﬁne a map g ′ = (gi′ )∞ i=1 by the following manner. Let D = ∞ ∪i=1 where D1 ⊂ D2 ⊂ . . . is a sequence of compact subsets, Bi = α−1 ([2−i , +∞)), Bi′ = Bi ∩ Di . for convenience, let B0 = B−1 = ∅. ′ Put gi = (g1′ , . . . , gi′ ) : C → Qi , then gi+1 = (gi , gi+1 ). ′ Suppose that the maps gi are deﬁned for each i < j and the following properties are satisﬁed for the maps gi : (a)i gi |(C\ int Bi−1 ) = fi |(C\ int Bi−1 ); ′ ) ⊂ Zi−1 , gi (Bi−1 \Di−1 ) ⊂ Σi \Zi−1 ; (b)i gi (Bi−2 (ci ) gi |Bi−2 is an embedding. Now we show how to construct a map gj′ so that the map gj satisﬁes the properties (a)j –(c)j . By property (3) of the map pjj−1 there exists an ′ embedding hj : Bj−2 → Zi−1 such that pjj−1 hj = gj−1 . Since Zj−1 is contained in a set of the form Qj−1 × K where K is a Z-set in Q, there exists a map r : Bj−2 → Q such that r(Bj−1 \Dj−1 ) ⊂ Q\K and the map (gj−1 |Bj−2 , r) : Bj−2 → Qj is an embedding. Now ﬁnd a map gj′ such that gj′ |(C\ int Bj−1 ) = πj fj |(C\ int Bj−1 ) (here πj : Qj → Q denotes the projection onto the last factor) and gj′ |Bj−2 = r. Obviously, the conditions (a)j - (c)j are satisﬁed. It is easy to see that g ′ |D = f |D. Show that the map g ′ = (gi′ )∞ i=1 is an embedding. First, let c ∈ C\D and α(c) ≥ 2−i for some i ∈ N. This means that c ∈ (C\ int Bi ) and, by property (a)i , ∞ ∑ d(f (c), g ′ (c)) = 2−j−1 = 2−i−1 , j=i+1

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CONSTRUCTIONS OF ABSORBING SPACES

whence g ′ (c) ∈ / f (D), by the property of α. If c1 , c2 ∈ C\D and c1 ̸= c2 , then c1 , c2 ∈ Bi for some i ∈ N. Then, by property (c)i+2 , gi+2 (c1 ) ̸= gi+2 (c2 ) and, consequently, g ′ (c1 ) ̸= g ′ (c2 ). Note also that, by the construction of g ′ , the set g ′ (C\D) is contained in ω Σ and, being σ-compact, it is a countable union of Z-sets. By well-known properties of Z-sets, the set g ′ (C) is a Z-set in Qω . We have to verify that (g ′ )−1 (X)\A = D\A. The inclusion ⊃ is obvious. To prove the inclusion ⊂ take c ∈ C\(D ∪ A). There exists i ∈ N such that c ∈ Bi \Di . The conditions (b)j for j > i imply g ′ (c) ∈ Σω \X. We have already proved that X is strongly C-universal in Qω . Note that X is contained in the countable union of sets of the form Q × ··· × Q × K × Q × ... where K are Z-sets in Q. Since these sets are Z-sets in Qω , the pair (Qω , X) is (M0 , C)-absorbing. Exercises and Problems to §2.5. 1. Express the weak product W (Xi , ∗i ) as the weak inverse limit of an inverse sequence. 2. Expand Theorem 2.5.9 to subclasses of C where C ∈ B. 3. (Open problem) Let (Q, X) be an (M0 , σM0 (na))-absorbing pair. Is X an ANRspace? 4. (Open problem) Find a counterpart of Theorem 2.5.9 for Γ-systems (see Ex.12 to §1.7). 5. Show that there exists a space X ∈ M1 [n], X ∈ AE(n) which is everywhere M1 [n]i+1 i universal. Hint: Take X = lim is a ←−{Xi , pj } where X1 = {∗}, Xi ∈ M1 [n], pi composition fi

pr

Xi+1 −−−−−→ Xi × s −−−−−→ Xi , fi is an n-soft map (see Theorem 2.3.33). 6. Show that the N¨ obeling space Nn is everywhere M1 [n]-universal.

§2.6. Some negative results The aim of this section is to collect some results on nonexistence of Cabsorbing spaces. 2.6.1. Theorem. For every countable limit ordinal ξ there is no M0 [ind, ξ]absorbing space. Proof. Assume ∪∞ the contrary and let X be an M0 [ind, ξ]-absorbing space. Then X = i=1 Xi for a sequence of compacta X1 , X2 , . . . with ind⊔Xi ≤ ξ. ∞ Denote by Z the one-point compactiﬁcation of the topological sum i=1 Xi . Obviously, Z ∈ M0 [ind, ξ]. Show that Z is M0 [ind, ξ]-universal.

HISTORICAL NOTES AND COMMENTS TO CHAPTER II

93

˜ ∈ M0 [ind, ξ] Let L ∈ M0 [ind, ξ]. By Theorem 2.2.18, there exists L ˜ satisfying the property: each nonempty open subset of L contains a topo˜ → X and, by the Baire logical copy of L. There exists an embedding f : L ˜ Category Theorem, some Xi ∩ f (L) contains a topological copy of L and hence Z contains a topological copy of L. Thus Z is M0 [ind, ξ]-universal and this contradicts to Theorem 2.2.51. The following result can be proved similarly using Theorem 2.2.52 instead Theorem 2.2.51. 2.6.2. Theorem. Let C ∈ B. For every countable limit ordinal α there is no C[dim, α]-absorbing space and C[Ind, α]-absorbing space. 2.6.3. Theorem. Let C ∈ {M0 , M1 , A1 }. There is no C(c.d.)-absorbing sets. Proof. By similarity, we consider only the case C = A1 . ∪ ∞ Assume that there exists A1 (c.d.)-universal space Y = i=1 Yi where Yi are compact countable-dimensional spaces. Then ind Yi = αi is deﬁned for each i ∈ N. Let α = sup{αi | i ∈ N} + 1. By Theorem 2.2.18 there exists a countable-dimensional compactum with the following property: every nonempty open subset of M contains a compactum Z with ind Z = α. Let f : M → Y be an embedding. By Baire Category Theorem, some f (M ) ∩ Yi contains a compactum Z with ind Z = α. This gives a contradiction. Problems to §2.6. 1. (Open problem) For which ordinal ξ there exists an M0 (ind, ξ)-absorbing space? 2. (Open problem) Are there M2 (c.d.)-absorbing spaces?

Historical Notes and Comments to Chapter II §2.2. Since in this book we deal with inﬁnite-dimensional spaces, we do not include in this survey the classical dimension theory concerning ﬁnitedimensional spaces. All deﬁnitions and results of classical theory the reader can ﬁnd in Engelking [1978] or Alexandrov, Pasynkov [1973] where he can also ﬁnd historical comments. The functions ind, Ind and dim are transﬁnite extensions of classical ﬁnite-valued functions. The transﬁnite dimension ind was introduced by Hurewicz [1928] and Ind by Smirnov [1959]. The function dim was recently introduced by Borst [1988]. The notion of (strong) countable-dimensionality was introduced by Hurewich [1928]. The notions of weak inﬁnite-dimensionality (A- or S-) were introduced by P.Alexandrov [1951] and Yu.Smirnov respectively.

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CONSTRUCTIONS OF ABSORBING SPACES

The majority of results of this section are well-known and one can ﬁnd them in the followings surveys: Engelking,Pol [1983], Chatyrko [1991] and Engelking [1980]. Theorem 2.2.26 is due to R.Pol [1981]. All notions and results from subsection E are due to V.Chatyrko [1993]. Theorem 2.2.50 is due to R.Pol [1986], Theorem 2.2.51 to W.Olszewski [1994] and Theorem 2.2.52 contains results from Olszewski [1994] and Radul,Zarichnyi [1996]. Theorem 2.2.53 is due to T.Dobrowolski and J.Mogilski [1990] and Proposition 2.2.56 to R.Pol [1983]. Finally, Theorems 2.2.57 and 2.2.58 are due to V.Chatyrko [1993]. §2.3. The notion of C-soft map was introduced by E.Shchepin [1981] and deeply investigated by A.Dranishnikov [1984] and A.Chigogidze [1986]. Theorem 2.3.3 is a modiﬁcation of Dranishnikov’s Factorization Theorem [1994]; its proof is based on the theory of inverse spectra due to E.Shchepin [1981]. Corollary 2.3.5 in a much stronger form was proved by A.Dranishnikov [1984]; Corollary 2.3.6 and Proposition 2.3.7 are due to M.Zarichnyi (see [1995a] for the latter). Lemma 2.3.8, Theorems 2.3.9, 2.3.10 and Corollary 2.3.11 are due to T.Radul. §2.4. Theorems 2.4.1—2.4.6 are due to T.Banakh and R.Cauty [199?a], Theorem 2.4.7 due to T.Dobrowolski, J.Mogilski [1990], Theorems 2.4.9 2.4.11 are due to M.Zarichnyi [1995b,1995a,199?]; see also T.Radul [1995] for Theorem 2.4.10. §2.5. The construction of Ωf is due to M.Zarichnyi. Theorem 2.5.5 is due to T.Radul and Theorem 2.5.9 is due to M.Zarichnyi. §2.6. Theorems 2.6.1 and 2.6.3 are due to M.Zarichnyi [1995] and Theorem 2.6.2 is proved by T.Radul and M.Zarichnyi [1996].

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Chapter III Advanced topics

In this chapter we discover and study the following phenomenon: the more complex and rich a class C is, the more interesting and deep theory of strongly C-universal and C-absorbing spaces can be developed. As a result, we know much less about strongly M0 -universal spaces comparing to strongly C-universal spaces, where C is any suﬃciently high Borel class. In its turn, for projective classes, we can prove even more striking results (e.g. that deleting an analytic subset from Π2 does not change its topological type). The main results of this chapter are Theorem 3.1.4 on equivalence of strong universality for pairs and for spaces, and Theorems 3.2.18–19 supplying us with an easy-to-apply characterization of strong C-universality for “nice” classes C. Now we will deﬁne some properties of classes of spaces, which will be exploited throughout the chapter (some of them have been deﬁned earlier). Let C, D be two classes of spaces. We deﬁne the classs C to be • T -stable, where T is a space, if C × T ∈ C for every space C ∈ C; • compactiﬁcation-admitting if for every space C ∈ C there exists a compactum K ∈ C containing C; • D-hereditary, if D(C) ⊂ C for every C ∈ C (for the deﬁnition of the class D(C) see §2.1); • D-additive, if every space X that can be expressed as X = C ∪ D, C ∈ C, D ∈ D, belongs to the class C; • weakly D-additive, if for every compactum K ∈ C and subsets C, D ⊂ K, C ∈ C, D ∈ D, we have C ∪ D ∈ C. All classes considered in this chapter are assumed to be topological and closed-hereditary. §3.1. Interplay between strongly universal pairs and strongly universal spaces The main result of this section is Theorem 3.1.3. In order to make the idea of its proof more transparent and understandable, we will prove ﬁrstly a corresponding result concerning universal spaces and universal pairs.

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ADVANCED TOPICS

A. Universal spaces and universal pairs. Obviously, if for a compactiﬁcation-admitting class of spaces C a pair (M, X) is (M0 ∩ C, C)-universal then the space X is C-universal. The following theorem shows that in some cases the converse is also true. 3.1.1. Theorem. Let X be a C-universal space for a 2ω -stable weakly A1 additive class C. Then for every Polish space M ⊃ X the pair (M, X) is (M0 ∩ C, C)-universal. Proof. Fix a pair (K, C) ∈ (M0 ∩C, C). Let A ∈ A1 \M1 be any dense subset in the Cantor cube 2ω , and consider the subset K × A ∪ C × 2ω ⊂ K × 2ω . Since the class C is 2ω -stable and weakly A1 -additive, K × A ∪ C × 2ω ∈ C. Let f : K × A ∪ C × 2ω → X be a closed embedding. By Lavrentiev Theorem, it extends to an embedding f¯ : G → M of some Gδ -set G ⊂ K × 2ω containing K × A ∪ C × 2ω . By Ex.10 to §1.1.A, (1)

f¯−1 (X) = K × A ∪ C × 2ω .

Notice that the complement (K × 2ω )\G is σ-compact and its projection B onto 2ω is a σ-compactum lying in 2ω \A. Since A ∈ A1 \M1 , we have 2ω \A ∈ M1 \A1 , and hence, B ̸= 2ω \A, i.e. there exists a point (2)

t ∈ 2ω \(A ∪ B).

Then K × {t} ⊂ G. Deﬁne the embedding e : K → M by e(k) = f¯(k, t), k ∈ K, and notice that it follows from (1) and (2) that e−1 (X) = C. 3.1.2. Corollary. If C is a 2ω -stable weakly A1 -additive compactiﬁcationadmitting class of spaces then a space X is C-universal if and only if it contains a C-universal Gδ -subset. Proof. The “only if” part is trivial. ˜ be any comAssume that G ⊂ X is a C-universal Gδ -subspace. Let X ˜ ⊂X ˜ be a Gδ -set such that G ˜ ∩ X = G. By Theorem pletion of X and G ˜ G) is (M0 ∩ C, C)-universal. Since G ˜ ∩ X = G, this im3.1.1, the pair (G, ˜ X) is (M0 ∩ C, C)-universal. Since the class C admits plies that the pair (X, compactiﬁcations, this yields X is C-universal. Exercises and Problems to §3.1.A.

1. Show that for any (M0 , C)-universal pair (M, X) the pair (M × s, X × s) is (M1 , C)universal. 2. Suppose C is a 2ω -stable weakly A1 -additive class such that for every C ∈ C there ˜ ∈ C. Show that for every embedding exists an everywhere C-universal Baire space C X ⊂ M of a C-universal space X into a space M ∈ M21 the pair (M, X) is (M0 ∩ C, C)universal. Hint: Use Theorem 3.1.1, Baire Theorem and the fact M21 = σM1 . 3. Suppose C is a 2ω -stable weakly A1 -additive compactiﬁcation-admitting class of spaces such that for every space C ∈ C there exists an everywhere C-universal Baire space

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˜ ∈ C. Show that a space X is C-universal if and only if X contains a C-universal C subset G ∈ M21 (X). 4. Suppose n ∈ N and C is a 2ω -stable weakly P2n−1 -additive class of spaces. Show that for every embedding X ⊂ M of a C-universal space X into a space M ∈ P2n the pair (M, X) is (M0 ∩ C, C)-universal. 5. Suppose n ∈ N and C is a 2ω -stable weakly P2n−1 -additive compactiﬁcation-admitting class of spaces. Show that a space X is C-universal if and only if it contains a Cuniversal subspace Y ∈ P2n (X). 6. Let X be a C-universal space, where C is an Nω -stable class of spaces. Show that for every σ-compact set A ⊂ X and every A′ ⊂ A the space X\A′ is C-universal. 7. Let X be a C-universal space, where C is a 2ω -stable weakly A1 -additive Gδ -hereditary compactiﬁcation-admitting class of spaces. Show that for every Fσ -set A ⊂ X belonging to the class M21 , and every A′ ⊂ A the space X\A′ is C-universal. 8. Show that a class C is M0 -hereditary (resp. M1 -hereditary) if and only if C is closedhereditary (resp. Gδ -hereditary). 9. Suppose n ∈ N, C is a 2ω -stable P2n -hereditary class of spaces, and X is a C-universal space. Show that for every subset A ∈ P2n−1 in X and any A′ ⊂ A the space X\A′ is C-universal. 10. (Open Problem) Let M be a space, X a subspace of M , 0 ≤ n ≤ ∞, and α ≥ 1 a countable ordinal. a) Suppose M ∈ Mα and X is an Aα [n]-universal space. Is the pair (M, X) (M0 [n], Aα )-universal? b) Suppose M ∈ Aα and X is an Mα [n]-universal space. Is the pair (M, X) (M0 [n], Mα )-universal? Remark that for n = 0 and α ≥ 2 the answer to b) is “yes”. (This follows from A.Kechris [1995, 28.19]).

B. Strong universality for spaces implies strong universality for pairs. Enlarging Theorem for strongly C-universal ANR’s. In this subsection we reverse Theorem 1.7.9 by proving “strongly universal” counterparts of the results of the previous subsection. 3.1.3. Theorem. Let M be a Polish ANR, X a homotopy dense subset in M , and let C be a 2ω -stable weakly A1 -additive class of spaces. If the space X is strongly C-universal then the pair (M, X) is strongly (M0 ∩ C, C)universal. Proof. Suppose X is strongly C-universal. To prove the strong (M0 ∩ C, C)-universality of the pair (M, X) ﬁx a pair (K, C) ∈ (M0 ∩ C, C), a closed subset B ⊂ K, a cover U ∈ cov(M ) and a map f : K → M whose restriction f |B : B → M is a Z-embedding with (f |B)−1 (X) = B ∩ C. Since f (B) is a Z-set in M and X is homotopy dense in M , replacing f by a close map, if necessary, we may assume that f (K\B) ⊂ X\f (B). Fix any metric d on M and let V ∈ cov(M ) be a cover such that V ≺ U and V ≺ {B(x, d(x, f (B))/2) | x ∈ M \f (B)}. Let A ∈ A1 \M1 be any dense set in 2ω and consider the subspaces C ′ = C × 2ω ∪ K\B × A and C ′′ = C\B × 2ω ∪ K\B × A in K × 2ω . Since the class C is 2ω -stable

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and weakly A1 -additive, C ′ ∈ C. Denote by prK : K × 2ω → K the natural projection and consider the map f ′ = f ◦ prK |C ′ : C ′ → X. Notice that (f ′ )−1 (X\f (B)) = C ′′ . By 1.5.10, there is a Z-embedding g : C ′′ → X\f (B) such that (1)

(g, f ′ |C ′ ) ≺ V.

Since g(C ′′ ) is a Z-set in X\f (B), ClM (g(C ′′ )) is a Z-set in M . By Lavrentiev Theorem, the embedding g extends to an embedding g¯ : G → M \f (B) of some Gδ -set G ⊂ (K\B)×2ω densely containing C ′′ . By Ex.10 to §1.1.A, (2)

g¯−1 (X\f (B)) = C ′′ .

Moreover, because of (1), we may suppose that (3)

(¯ g , f ′ |G) ≺ V ≺ U.

Remark that the complement (K\B × 2ω )\G is σ-compact and its projection P = pr((K\B × 2ω )\G) onto 2ω is a σ-compactum in 2ω \A ∈ / A1 . Then there is a point t ∈ 2ω \(A ∪ P ). Notice that K\B × {t} ⊂ G and deﬁne the map f˜ : K\B → M \f (B) letting f˜(k) = g¯(k, t) for k ∈ K\B. By (2) and (3) we have f˜−1 (X\f (B)) = K\C and (f˜, f |K\B) ≺ U. Letting ﬁnally f¯ : K → M be the map deﬁned by f¯|B = f |B and f¯|K\B = f˜|K\B, we obtain a closed embedding f¯ such that f¯−1 (X) = C and (f¯, f ) ≺ U. To see that f¯(K) is a Z-set in M , notice that f¯(K) ⊂ f (B)∪ClM (g(C ′′ )) lies in the union of two Z-sets in M . Theorems 3.1.3 and 1.7.9 immediately imply 3.1.4. Theorem. Let C be a 2ω -stable weakly A1 -additive compactiﬁcationadmitting class of spaces, M a Polish ANR and X ⊂ M a homotopy dense subspace satisfying SDAP. The space X is strongly C-universal if and only if the pair (M, X) is strongly (M0 ∩ C, C)-universal. We apply Theorem 3.1.4 to prove 3.1.5. Enlarging Theorem. Let C be a 2ω -stable weakly A1 -additive compactiﬁcation-admitting class of spaces. An ANR X satisfying SDAP is strongly C-universal if and only if X contains a strongly C-universal homotopy dense Gδ -subspace G ⊂ X. Proof. Suppose G is a strongly C-universal homotopy dense Gδ -subspace of X. Using Theorem 1.2.4, ﬁnd a Polish ANR M containing X as a homotopy ˜ ⊂ M be any Gδ -subset with G ˜ ∩ X = G. Then G dense subset. Let G is homotopy dense in M (see Ex.7 to §1.2). By Theorem 3.1.4, the pair ˜ G) is strongly (M0 ∩ C, C)-universal. Since G ˜ is homotopy dense in M , (G,

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we can apply Theorem 1.7.10, to prove that the pair (M, X) is strongly (M0 ∩ C, C)-universal. Finally, by Theorem 1.7.9, the space X is strongly C-universal. Finally, let us prove a characterization of the strong C-universality, which is related to Proposition 1.5.7. 3.1.6. Theorem. Let C be an Nω -stable open-hereditary class of spaces and let X be an ANR such that every Z-set in X is strong. The space X is strongly C-universal if and only if for every open set U ⊂ X, every map f : C → U of a space C ∈ C can be approximated by a closed embedding f¯ : C → U . Proof. The “only if” part follows from 1.5.6. Suppose that X satisﬁes the hypothesis of the “if” part. To show that X is strongly C-universal we will apply 1.5.7. Fix an open set U ⊂ X, a cover U ∈ cov(U ), and a map f : C → U of a space C ∈ C. By the hypothesis, there is a closed embedding g : C × Nω → U such that (g, f ◦ prC ) ≺ U, where prC : C × Nω → C stands for the projection. Let ∪A be a countable dense subset in the function space C(Q, U ) and let A = α∈A α(Q) ⊂ U . Evidently, the set A is σ-compact, and thus g −1 (A) is a σ-compact set in C × Nω . Then the projection P of g −1 (A) onto Nω is σ-compact as well, and thus we can pick a point t0 ∈ Nω \P . Clearly, C × {t0 } ∩ g −1 (A) = ∅. Deﬁne the embedding f¯ : C → U by f¯(c) = g(c, t0 ) for c ∈ C. Evidently, f¯ is a closed embedding and f¯(C) ∩ A = ∅. Then C(Q, U \f¯(C)) ⊃ A is dense in C(Q, U ) and thus f¯(C) is a Z∞ -set. By 1.4.4, f¯(C) is a Z-set in U . Exercises and Problems to §3.1.B. 1. Suppose n ∈ N, C is a 2ω -stable weakly P2n−1 -additive

class of spaces, M ∈ P2n is an ANR, and X is a homotopy dense strongly C-universal subspace in M . Show that the pair (M, X) is strongly (M0 ∩ C, C)-universal.

2. Suppose n ∈ N, C is a 2ω -stable weakly P2n−1 -additive compactiﬁcation-admitting class of spaces, M ∈ P2n is an ANR and X ⊂ M be a homotopy dense subspace satisfying SDAP. Show that the space X is strongly C-universal if and only if the pair (M, X) is strongly (M0 ∩ C, C)-universal.

3. Suppose n ∈ N and C is a 2ω -stable weakly P2n−1 -additive compactiﬁcation-admitting class of spaces. An ANR X satisfying SDAP is strongly C-universal if and only if it contains a strongly C-universal homotopy dense subspace G ∈ P2n (X). 4. Let C, D be two classes of spaces. Suppose, there is a space C ∈ / Dσ such that (i) I n × C ∈ C for every n ∈ N and (ii) a subset D ⊂ C belongs to the class Dσ whenever there exists a (perfect) surjective map f : D ′ → D, where D′ ∈ D. Show that for a strongly C-universal ANR X any (Fσ -)subset F ∈ Dσ in X is homotopy negligible. 5. Suppose X is a strongly C-universal ANR satisfying SDAP, where C ⊃ {I n | n ∈ N} is an Nω -stable class of spaces. Show that for every σ-compact set A ⊂ X and

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every A′ ⊂ A (i) A is homotopy negligible in X and (ii) the space X\A′ is strongly C-universal.

6. Suppose n ∈ N, C ⊃ {I k | k ∈ N} is a 2ω -stable P2n -hereditary class of spaces, and X is a strongly universal ANR satisfying SDAP. Show that for every set A ∈ P2n−1 in X and every A′ ⊂ A (i) A is homotopy negligible in X and (ii) the space X\A′ is strongly C-universal. A subset A of a space X is deﬁned to be strongly negligible in X, provided for every cover U ∈ cov(X) there is a homeomorphism h : X → X\A, U -close to id. The following are counterparts of Strong Negligibility Theorem 1.1.19. 7. Let C ⊃ {I n | n ∈ N} be a 2ω -stable Gδ -hereditary class of spaces, and let Ω be a C-absorbing space. Show that every σ-compact subset of Ω is strongly negligible in Ω. 8. Let n ∈ N, C ⊃ {I k | k ∈ N} be a 2ω -stable P2n -hereditary class of spaces, and let Ω be a C-absorbing space. Show that every subset in Ω of the class P2n−1 is strongly negligible in Ω. 9. (Open Problem) Is every subset of the class Aα (resp. Mα ) strongly negligible in Ωα (resp. Λα )?

C. Homotopy dense embeddings of absorbing spaces. In this subsection we deal with the question: given an absorbing space Ω, how much topologically distinct pairs (Q, X) with X a homotopy dense subset of Q, homeomorphic to Ω, does exist? We start with 3.1.7. Proposition. Let Ω be a C-absorbing AR, where C ⊂ A1 is a closed-hereditary topological class of σ-compacta. Then (1) there exists a homotopy dense embedding Ω ⊂ s such that the pairs (s, Ω) and (Q, Ω) are absorbing; (2) a pair (Q, X) is homeomorphic to (Q, Ω) if and only if X is homotopy dense in Q and X is homeomorphic to Ω; (3) a pair (s, X) is homeomorphic to (s, Ω) if and only if X is homotopy dense in s and X is homeomorphic to Ω. Proof. By Theorem 1.3.2, there is a homotopy dense embedding Ω ⊂ s. Let us show that the pairs (s, Ω) and (Q, Ω) are absorbing. One can easily deduce from strong C-universality of Ω that the pairs (s, Ω) and (Q, Ω) are ⃗ strongly C-universal, where C⃗ = {(K, K) | K ∈ C ∩ M0 }. Since Ω ∈ σC and ⃗ Finally, since Ω is a σ-compact C ⊂ A1 , we can show that (Ω, Ω) ∈ σ C. space, Ω is a Zσ -set in s as well as in Q. Thus the pairs (s, Ω) and (Q, Ω) ⃗ are C-absorbing, and, by Theorem 1.7.8, they are absorbing. Therefore, the statement 1) is proved. 2) If X is a homotopy dense Zσ -subspace of Q, homeomorphic to Ω, then ⃗ by the above arguments, we can prove that the pair (Q, X) is C-absorbing and by Uniqueness Theorem 1.7.6, the pair (Q, X) is homeomorphic to (Q, Ω).

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The third statement can be proved by analogy with the previous one. If C ̸⊂ A1 then the situation changes. The statements 1),2) still hold for certain “nice” classes C, while 3) does not (see exercises to this subsection). 3.1.8. Theorem. If Ω is a C-absorbing AR for an I-stable class C then there is a homotopy dense embedding Ω ⊂ s such that Ω lies in a σ-compact subset A ⊂ s and the pairs (s, Ω), (Q, Ω) are absorbing. Moreover, if the class C admits compactiﬁcations then we can assume that A ∈ σ-(M0 ∩ C). Proof. We will modify a little bit the construction from §2.4. Let Ω be a C∪∞ absorbing AR, where C is an I-stable class. Write Ω = n=1 Cn , where Cn ∈ C, n ∈ N, are closed subsets in Ω. In the case if C admits compactiﬁcations, ﬁnd for every n ∈ N a compactum Kn ∈ C with Cn ⊂ Kn (in the general case, let Kn be any compactiﬁcation of Cn ). Let K = ⊔n∈N Kn , C = ⊔n∈N Cn and C⃗ = ⊔n,m∈N F0 (Kn × I m , Cn × I m ). Analogously as in §2.4, we consider K × [0, 1] as a subset of a linearly independent compactum in l2 . Let D be any countable dense subset in K × (0, 1]. Analogously as in 2.4.1, we can assume that span D is dense in l2 . Consider the subsets Ω(K) = {tx + y | t ∈ I, x ∈ K, y ∈ span D} ⊂ l2 , Ω(C) = {tx + y | t ∈ I, x ∈ C, y ∈ span D} ⊂ l2 . To prove the theorem, it is enough to establish the following facts: 1) Ω(C) is homeomorphic to Ω, 2) the pair (l2 , Ω(C)) is absorbing, 3) identifying l2 with s ⊂ Q (by Anderson-Kadec Theorem 1.1.27), we obtain an absorbing pair (Q, Ω(C)), and 4) the set A = Ω(K) belongs to the class σ-(M0 ∩ C), if the class C admits compactiﬁcations. Analogously as in the proof of 2.4.1 we can show that ⃗ a) the pair (l2 , Ω(C)) is strongly C-universal; ⃗ b) (Ω(K), Ω(C)) ∈ σ C; c) Ω(C) is a homotopy dense subset in l2 , and thus Ω(C) is an AR with SDAP; d) Ω(K) is a σ-compact subset in l2 and thus both Ω(K) and Ω(C) are strong Zσ -spaces. Using a), c), and applying Theorem 1.7.9, we get the space Ω(C) is strongly Cn -universal for every n∪ ∈ N. Since Ω(C) is a strong Zσ∪-space, by ∞ ∞ Theorem 1.5.8, Ω(C) is strongly n=1 Cn -universal. Since Ω = n=1 Cn is a C-universal space, we get Ω(C) is a strongly C-universal space. It follows from b) and I-stability of C that Ω(C) ∈ σC. Then Ω(C) is a C-absorbing AR, and by Theorem 1.6.3, Ω(C) is homeomorphic to Ω. Therefore, the statement 1) is proved. ⃗ 2) Since the pair (Ω(K), Ω(C)) is strongly C-universal and Ω(K) is ho2 motopy dense in l , we can apply Theorem 1.7.10 to prove that the pair

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⃗ (l2 , Ω(C)) is strongly C-universal. Because Ω(K) is a σ-compact (and thus 2 ⃗ we obtain that (l2 , Ω(C)) is a a Zσ -) set in l with (Ω(K), Ω(C)) ∈ σ C, ⃗ C-absorbing pair, and by 1.7.8, this pair is absorbing. Thus 2) holds. By the same argument, we can prove the statement 3). Finally, it follows from b) that the set Ω(K) belongs to the class σ-(M0 ∩ C), whenever the class C admits compactiﬁcations. ∼ Ω and X ⊂ A ⊂ s, where A ∈ σ-(M0 ∩ C) can be Pairs (s, X) with X = considered as minimal in some sense. The following Theorem shows that all such pairs are homeomorphic. 3.1.9. Theorem. Suppose C is a 2ω -stable weakly A1 -additive compactiﬁcation-admitting class, and M is either a Q-manifold or an s-manifold. For i = 1, 2, let Xi ⊂ M be a C-absorbing homotopy dense subspace contained in a subset Ai ⊂ M belonging to the class σ-(M0 ∩ C). Then the pairs (M, X1 ) and (M, X2 ) are homeomorphic. Proof. Let i ∈ {1, 2}. Applying Theorem 3.1.3, we obtain that the pair (M, Xi ) is strongly (M0 ∩C, C)-universal. Existence of the set Ai ∈ σ-(M0 ∩ C) with Xi ⊂ Ai implies that (A′i , Xi ) ∈ σ-(M0 ∩ C, C) for some subset A′i such that Xi ⊂ A′i ⊂ Ai . Moreover, since Xi is a homotopy dense Zσ subspace in M , we can assume that A′i is a Zσ -set in M . Thus the pair (M, Xi ) is (M0 ∩ C, C)-absorbing. By Theorem 1.7.6, the pairs (M, X1 ) and (M, X2 ) are homeomorphic. Now we have come to the main result of this subsection. 3.1.10. Theorem. Let either C ⊂ A1 or C be a 2ω -stable A1 -additive class of spaces, and let Ω be a C-absorbing AR. Then (1) there exists a homotopy dense embedding Ω ⊂ s such that the pairs (s, Ω), (Q, Ω) are (M0 , C)-absorbing; (2) a pair (Q, X) is homeomorphic to (Q, Ω) if and only if X is a homotopy dense subset of Q, homeomorphic to Ω; (3) a pair (s, X) is homeomorphic to (s, Ω) if and only if X is a homotopy dense subset of Q, homeomorphic to Ω and lying in a σ-compact subset of s. Proof. If C ⊂ A1 , our theorem results from Proposition 3.1.7. So we assume that C is 2ω -stable A1 -additive class of spaces. By Ex.7 to §1.3, there exists a homotopy dense embedding of Ω into the Hilbert cube. Since Ω is a Zσ space, there is a Zσ -set Z ⊂ Q with Ω ⊂ Z. Using Ex.8 to §1.7, we can construct a homeomorphism h : Q → Q such that h(Z) ∩ Bd(Q) = ∅. Thus, without loss of generality, we can suppose Z ⊂ s. It follows from Theorem 3.1.3 that the pairs (s, Ω) and (Q, Ω) are (M0 , C)-absorbing. Thus the statement 1) is proved. The statements 2) and 3) follow from Theorem 3.1.9.

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3.1.11. Remark. Clearly, all the Borel classes Mα , Aα excepting M0 and M1 , and all the projective classes Pn satisfy the assumptions of Theorem 3.1.10. Exercises and Problems to §3.1.C. 1. Consider the pairs (Q×Q, s×σ) and (Q×Q, (Q\σ)×σ) and show that (i) the sets s×σ and (Q\σ) × σ are homotopy dense in Q × Q, (ii) the spaces (Q\σ) × σ and s × σ are homeomorphic, and (iii) the complements (Q × Q)\(s × σ) and (Q × Q)\(Q\σ × σ) are not homeomorphic (and consequently the pairs (Q × Q, s × σ) and (Q × Q, (Q\σ) × σ) are not homeomorphic neither). 2. For i = 1, 2, let Ci be an I-stable class of spaces and Ωi a Ci -absorbing homotopy dense subspace in Q. Show that if the complements of (Q\Ω1 ) × σ and (Q\Ω2 ) × σ in Q × Q are homeomorphic then Ω1 and Ω2 are homeomorphic as well. By c we denote the continuum cardinal. In Ex.3–14 we suppose that Ω is a C-absorbing AR for a topological I-stable closed-hereditary class C. 3. Construct a collection {Xα }α∈c of homotopy dense subsets of Q such that (i) for every α ∈ c the pair (Q, Xα ) is absorbing, and Xα is homeomorphic to s × σ; (ii) for distinct α, β ∈ c the complements Q\Xα , Q\Xβ are not homeomorphic (and consequently, the pairs (Q, Xα ), (Q, Xβ ) are not homeomorphic). 4. Suppose the class C admits compactiﬁcations and contains the space

Nω .

a) Assuming that C does not contain the Hilbert cube, construct two homotopy dense subsets X0 , X∞ ⊂ Q such that the pairs (Q, X0 ), (Q, X∞ ) are absorbing, X0 , X∞ are homeomorphic to Ω, but the complements Q\X0 , Q\X∞ are not homeomorphic (and thus the pairs (Q, X0 ), (Q, X∞ ) are not homeomorphic). b) Assuming that C contains no strongly inﬁnite-dimensional compactum, construct a collection {Xα }α∈c of homotopy dense subsets in Q such that (i) for every α ∈ c the pair (Q, Xα ) is absorbing and Xα is homeomorphic to Ω and (ii) for distinct α, β ∈ c the complements Q\Xα , Q\Xβ are not homeomorphic (and thus the pairs (Q, Xα ), (Q, Xβ ) are not homeomorphic). 5. Assuming that Nω ⊂ C, construct two homotopy dense subsets X0 , X∞ ⊂ s such that (i) the spaces X0 , X∞ are homeomorphic to Ω, (ii) the pairs (s, X0 ) and (s, X∞ ) are absorbing, but (iii) the pairs (s, X0 ) and (s, X∞ ) are not homeomorphic. 6. Assuming that M1 ⊂ C, construct a collection {Xα }α∈c of homotopy dense subsets in s such that (i) for every α ∈ c the space Xα is homeomorphic to Ω and the pair (s, Xα ) is absorbing and (ii) the pairs (s, Xα ) are pairwise topologically distinct. 7. Suppose s contains two topological copies X1 , X2 of Ω such that (s, X1 ), (s, X2 ) are topologically non-equivalent strongly universal pairs. Construct a collection {Xα }α∈c of homotopy dense subsets in s such that (i) the spaces Xα are homeomorphic to Ω, (ii) the pairs (s, Xα ) are not strongly universal and (iii) the pairs (s, Xα ) ∪ are pairwise ∞ topologically distinct. Hint: Consider the subspace T = [0, ∞) × {0} ∪ n=1 {n} × 2 [0, 1] ⊂ R and for every function α : N → {0, 1} let Xα =

∞ ∪

(2n − 2, 2n − 1) × {0} × X1 ∪ (2n − 1, 2n) × {0} × X2 ∪ {n} × [0, 1] × Xα(n) .

n=1

Show that the sets Xα ’s in T × s are as required. 8. Suppose Q contains two topological copies X1 , X2 of Ω such that (Q, X1 ), (Q, X2 ) are topologically non-equivalent strongly universal pairs. Construct a collection {Xα }α∈c

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of homotopy dense subsets in Q such that (i) the spaces Xα are homeomorphic to Ω, (ii) the pairs (Q, Xα ) are not strongly universal, and (iii) the pairs (Q, Xα ) are pairwise topologically distinct. Hint: Denote by αT the one-point compactiﬁcation of the space T from the previous exercise and show that the sets Xα in αT × Q are as required. 9. (Open Problem) Suppose X is a homotopy dense Fσ -subset in s, homeomorphic to s × σ. Is the pair (s, X) homeomorphic to (s × s, s × σ)? 10. (Open Problem) Let C be a 2ω -stable A1 -additive class of spaces. Do there exist two topologically distinct pairs (Q, X), (Q, Y ), where X, Y ⊂ Q are homotopy dense C(c.d.)-absorbing subspaces in Q? 11. (Open Problem) For which classes C there is an embedding Ω ⊂ Q such that the pair (Q, Ω) is (M0 , C)-absorbing? 12. Show that an embedding Ω ⊂ Q such that the pair (Q, Ω) is (M0 , C)-absorbing exists if and only if the class (M0 , σC) contains an (M0 , C)-universal pair. 13. Suppose the class C is M1 -hereditary and the class (M0 , σC) contains an (M0 , C(f.d.))universal pair. Show that for the class C(f.d.) there is a C(f.d.)-absorbing AR Ω and an embedding Ω ⊂ Q such that the pair (Q, Ω) is strongly (M0 , C(f.d.))-universal. 14. Suppose the class C is A2 -hereditary and the class (M0 , σC) contains an (M0 , C(c.d.))universal pair. Show that for the class C(c.d.) there is a C(c.d.)-absorbing AR Ω and an embedding Ω ⊂ Q such that the pair (Q, Ω) is strongly (M0 , C(c.d.))-universal.

D. Applications to coabsorbing spaces. We start with the following analog of Theorem 3.1.10. 3.1.12. Theorem. Let C be a 2ω -stable A1 -additive class, and Ω a Ccoabsorbing AR. Then (1) there exists a homotopy dense embedding Ω ⊂ Q such that the pair ⃗ (Q, Q\Ω) is C-absorbing, where C⃗ = {(K, C) | (K, K\C) ∈ (M0 , C)}; (2) a pair (Q, X) is homeomorphic to (Q, Ω) if and only if X is a homotopy dense subset in Q, homeomorphic to Ω. Proof. By Ex.7 to §1.3, there is a homotopy dense embedding Ω ⊂ Q. We ⃗ claim that the pair (Q, Q\Ω) is C-absorbing. By Theorem 3.1.3, the pair (Q, Ω) is strongly (M0 , C)-universal. Of course, this is equivalent to strong ⃗ C-universality of the pair (Q, Q\Ω). Since the space Ω is C-coabsorbing, ∪∞Ω contains a homotopy dense absolute Gδ -subset G ⊂ Ω. Write Q\G = n=1 Zn , where each Zn is closed in Q. Since G is homotopy dense in Q, each Zn is a Z-set in Q. Then Zn ∩ Ω is a Z-set in Ω, and hence Zn ∩ Ω ∈ C.This implies (Zn , Zn \Ω) ∈ C⃗ ∪∞ immediately ∪∞ and thus the pair (Q\G, Q\Ω) = ( n=1 Zn , n=1 Zn \Ω) belongs to the class ⃗ Since Q\G is a Zσ -set in Q, we get the pair (Q, Q\Ω) is C-absorbing. ⃗ σ-C. If X is any homotopy dense subset in Q, homeomorphic to Ω, then by ⃗ the same arguments as above, we can show that the pair (Q, Q\X) is C-

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absorbing and according to Theorem 1.7.6, this pair is homeomorphic to (Q, Q\Ω). Passing to the complements, we get the pairs (Q, X) and (Q, Ω) are homeomorphic. For a class C denote by C • the class of spaces C for which there is a compactum K containing C so that K\C ∈ C. The following theorem reveals duality between absorbing and coabsorbing spaces (cf. Ex.9 to §1.7). 3.1.13. Theorem. Let C be a 2ω -stable A1 -additive class of spaces such that σC = C. Suppose M is a Polish ANR with LCAP and X is a homotopy dense subspace in M . (1) If the space X is strongly C-universal then its complement M \X is strongly C • -universal. (2) If X is C-absorbing then M \X is C • -coabsorbing. (3) If X is C-coabsorbing then M \X is C • -absorbing. Proof. Suppose X is a strongly C-universal space. Then by Theorem 3.1.3, the pair (M, X) is strongly (M0 , C)-universal. Since X is homotopy dense in M , its complement M \X is homotopy negligible in M . In fact, M \X is also homotopy dense in M . Indeed, since M0 ⊂ C (C is A1 -additive!), the pair (M, X) is strongly (Q, ∅)-universal. By Ex.3 to §1.7, this implies the set X is homotopy negligible in M , and thus M \X is homotopy dense (and homotopy negligible) in M . Applying 1.2.1 and Ex.12h to §1.3, we obtain that M \X is an ANR with SDAP. Strong (M0 , C)-universality of the pair ⃗ (M, X) is equivalent to strong C-universality of the pair (M, M \X), where ⃗ C = {(K, C) | (K, K\C) ∈ (M0 , C)}. Applying Theorem 1.7.9, we get the space M \X is strongly C • -universal. Suppose now that X is a C-absorbing space. Obviously, the class C • is closed-hereditary. Thus to prove that each Z-set in M \X belongs to C • , it is enough to show that M \X ∈ C • . This results from the following Claim. For every Polish space A and a subspace C ⊂ A if C ∈ C then A\C ∈ C • . ¯ Proof. Let A¯ be any compactiﬁcation of A. Then A\A is a σ-compact set ¯ ¯ in A, and by A1 -additivity of C, we have A\A ∪ C ∈ C. By the deﬁnition of ¯ A\A ¯ ∪ C) ∈ C • . The claim is proved. C • , A\C = A\( Now let us show that M \X is a co-Zσ -space. Find a Zσ -set Z ⊂ M containing X (recall that X is a homotopy dense Zσ -subspace in M ). By Ex.3 to §1.4, Z is homotopy negligible in M . Then G = M \Z is a homotopy dense absolute Gδ -set in X. Thus the space M \X ∈ C • , being a strongly C • -universal co-Zσ -ANR, is a C • -coabsorbing space. Now assume that X is a C-coabsorbing space. Let G ⊂ X be a homotopy dense absolute Gδ -set in X. Then M \G is a Zσ -set in M . Since M \X ⊂

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M \G is homotopy dense in M , this implies M \X is a Zσ -space (see ∪∞Ex.16 to §1.4). Let us show ﬁnally that M \X ∈ σC • . Write M \G = n=1 Zn , where each Zn is a Z-set in M . Then Zn ∩ X is a Z-set∪in X, and thus ∞ Zn ∩ X ∈ C. By Claim, Zn \X ∈ C • . Therefore, M \X = n=1 Zn \X and each Zn \X ∈ C • is a closed set in M \X, i.e., M \X ∈ σC • . Exercises to §3.1.D. 1. Assuming that the class C in Ex.5 and 6 of §3.1.C is A1 -additive, show that the complements s\Ω0 , s\Ω∞ , Q\Ωα , s\Ωα , α ∈ c, are homeomorphic. Hint: Apply Theorems 1.6.4 and 3.1.13. For a class C let C ◦ be the class of spaces C such that for every compactum K ⊃ C we have K\C ∈ C. 2. Show that for every countable ordinal α ≥ 1 and every n ∈ N we have: M•α = M◦α = • ◦ • = P◦ = P • Aα , A•α = A◦α = Mα , P2n−1 = P2n−1 = P2n , P2n 2n−1 ; A0 = M0 , 2n A◦0 = ∅, M◦0 = A0 = ∅, and M•0 is the class of locally compact spaces. 3. Show that (C ◦ )◦ ⊂ C ⊂ (C • )• for every topological class C. 4. Suppose C is an M1 -hereditary A1 -additive topological class of spaces. Show that C◦ = C•. 5. Find a class C ̸= A1 such that C • = M1 . Hint: Consider the class C = A1 (s.c.d.). 6. For a class of spaces C prove the following statements: a) if C is T -stable, where T is a compactum, then C • is T -stable; b) if C is M0 -hereditary, then the classes C • and C ◦ are M0 -hereditary; c) if C is D-hereditary, where D is a class of spaces, then the classes C • and C ◦ are D◦ -additive. d) if C is D-additive, where D is a class of spaces, then the classes C • and C ◦ are D◦ -hereditary.

§3.2. Characterizing (strong) C-universality for “nice” classes C In this section we prove that for some “nice” classes C, an ANR X with SDAP is strongly C-universal if and only if the subset of all perfect maps is dense in C(C, U ) for every C ∈ C and open U ⊂ X, cf. 1.5.7 and 3.1.6. The proof is technically complicated and requires a great deal of preliminary work. A. Some properties of function spaces. Recall that for spaces X, Y by C(X, Y ) we denote the space of all continuous functions X → Y , endowed with the limitation topology whose neighborhood base at an f ∈ C(X, Y ) consists of the sets B(f, U) = {g ∈ C(X, Y ) | (g, f ) ≺ U}, where U runs over the set cov(Y ). For a closed subset Y ⊂ Z we identify the space C(X, Y ) with the subspace {f ∈ C(X, Z) | f (X) ⊂ Y } of C(X, Z). As we have already mentioned in §1.1, in the case of compact X, the space C(X, Y ) is separable and metrizable (an admissible metric for C(X, Y ) can

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ˆ g) = sup be deﬁned as d(f, x∈X d(f (x), g(x)), where d is a metric on Y ). Moreover, if d is complete, then dˆ is a complete metric on C(X, Y ). Therefore, if Y is Polish and X compact, then C(X, Y ) is a Polish space, and thus a Baire space. 3.2.1. Proposition. For every space X and every Polish space Y the function space C(X, Y ) is Baire. Proof. Let {Un }n∈N be ∩a countable collection of open dense sets in C(X, Y ). We should prove that n∈N Un is dense in C(X, Y ). Fix a complete metric d on Y , an f ∈ C(X, Y ), and a cover U ∈ cov(Y ). Let U0 ∈ cov(Y ) be a cover such that St U0 ≺ U and mesh U0 < 1 (recall that for a cover V ∈ cov(Y ) we set V¯ = {V¯ | V ∈ V}). Using density of the set U1 in C(X, Y ), pick up a function f1 ∈ U1 such that (f1 , f0 ) ≺ U0 . Since the set U1 is open in C(X, Y ), there is a cover U1 ∈ cov(Y ) such that B(f1 , U¯1 ) ⊂ U1 . We may assume that St U1 ≺ U0 and mesh U1 < 12 . Proceeding in this way, we will construct inductively a sequence of maps {fn : fn ∈ Un } and a sequence of covers {Un : Un ∈ cov(Y )} satisfying for every n the conditions: (1n ) B(fn , U¯n ) ⊂ Un , (2n ) (fn , fn−1 ) ≺ Un−1 , (3n ) St Un ≺ Un−1 , (4n ) mesh Un < 2−n . Because of (2n ), (4n ), n ∈ N, the sequence of maps {fn } is Cauchy, and hence converges to a function f ∈ C(X, Y ) (see Ex.8 to §1.1.A). Next, because of (1n ), (2n ), and (3n ), n ∈∩ N, we have (f, f0 ) ≺ U¯ and f ∈ ¯ ¯ B(f ∩ n , Un ) ⊂ Un for every n, i.e. f ∈ n∈N Un is U-close to f0 . Therefore, n∈N Un is dense in C(X, Y ). Recall that a map f : X → Y is called a U-map, where U ∈ cov(X), if there is a cover V ∈ cov(Y ) such that f −1 V ≺ U. 3.2.2. Proposition. a) For each U ∈ cov(X) the set of all U-maps is open in C(X, Y ). b) Let d be a complete metric on X and for every n ∈ N let Un ∈ cov(X) be such that mesh Un < n1 . Then, f : X → Y is a closed embedding if and only if f is a Un -map for every n ∈ N. Proof. To see a), ﬁx a U-map f : X → Y and ﬁnd a cover V ∈ cov(Y ) such that f −1 V ≺ U. Next, taking any cover W ∈ cov(Y ) with St W ≺ V, show that every map g : X → Y , W-close to f , is a U-map (in fact, g −1 W ≺ U). To see b), notice that a sequence {xn } ⊂ X is Cauchy, whenever {f (xn )} converges and f is a Un -map for all n. Proposition 3.2.2 and Theorem 1.1.21 imply

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3.2.3. Proposition. For every Polish space X and every s-manifold M , the set of all closed embeddings X → M is a dense Gδ -subset in C(X, M ). Exercises to §3.2.A. 1. Show that for every locally compact space X the set of all perfect maps is open in C(X, Y ). 2. Show that for every map (closed embedding) p : Y → Z the map C(X, Y ) → C(X, Z) sending an f ∈ C(X, Y ) onto p ◦ f ∈ C(X, Z) is continuous (is a closed embedding). 3. Show that for every compactum K and every space X the map C(K, X) → C(K, K × X), sending a map f : K → X onto the map idK × f : K → K × X, is continuous. 4. Let U ∈ cov(X). Show that a closed map f : X → Y is a U -map if and only if {f −1 (y)}y∈Y ≺ U .

B. Perfect multivalued maps. For a space X by exp(X) we denote the set of all compact subspaces of X (there is no topology on exp(X)!). Any function F : Z → exp(X) will be called a multivalued map. For subsets A ⊂ X and B ⊂ Z let F −1 (A) = {z ∈ Z | F(z) ∩ A ̸= ∅}, ∪ F(B) = F(z). z∈B

A multivalued map F : Z → exp(X) is called upper semicontinuous if for every closed set A ⊂ X the set F −1 (A) is closed in Z. It is easily seen that a multivalued map F : Z → exp(X) is upper semicontinuous if and only if for every z ∈ Z and a neighborhood U ⊂ X of F(z) there is a neighborhood V ⊂ Z of z such that F(V ) ⊂ U . A multivalued map F : Z → exp(X) is deﬁned to be perfect if for compact sets A ⊂ X and B ⊂ Z the sets F −1 (A) ⊂ Z and F(B) ⊂ X are compact. For two perfect multivalued maps F1 : Z → exp(Y ) and F2 : Y → exp(X) deﬁne their composition F2 ◦ F1 : Z → exp(X) by F2 ◦ F1 (z) = F2 (F1 (z)), z ∈ Z. Obviously, the composition F2 ◦ F1 is a perfect multivalued map. 3.2.4. Proposition. Any perfect multivalued map F : Z → exp(X) is upper semicontinuous. Proof. Fix a closed set F ⊂ X. Since the space Z, being metrizable, is a kspace, to show that F −1 (F ) is closed in Z, it suﬃces to prove that for every compactum K ⊂ Z the intersection K ∩ F −1 (F ) is compact. This follows from perfectness of F and the equality K ∩ F −1 (F ) = K ∩ F −1 (F(K) ∩ F ).

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3.2.5. Proposition. Let F : Z → exp(X) be a perfect multivalued map and U ∈ cov(X) a cover such that {F(z)}z∈Z ≺ U . Then there is a cover V ∈ cov(X) such that {St (F(z), V)}z∈Z ≺ U. Proof. Fix any metric d on the space X. Let z ∈ Z be any point. Find U (z) ∈ U with F(z) ⊂ U (z). Since F(z) is compact, there is δ(z) > 0 such that B(F(z), 2δ(z)) ⊂ U (z). Since the map F is upper semicontinuous, there is a neighborhood W (z) ⊂ Z of z such that F(W (z)) ⊂ B(F(z), δ(z)). Evidently, B(F(W (z)), δ(z)) ⊂ B(B(F(z), δ(z)), δ(z)) ⊂ B(F(z), 2δ(z)) ⊂ U (z). Let W ∈ cov(Z) be a locally ﬁnite cover of Z inscribed into the cover {W (z)}z∈Z . For every W ∈ W ﬁx a z ∈ Z with W ⊂ W (z) and let δ(W ) = δ(z). Then {B(F(W ), δ(W ))}W ∈W ≺ U. Now for every point x ∈ X we shall ﬁnd a neighborhood V (x) ⊂ X of x as follows. Since F −1 (x) is compact and the cover W is locally ﬁnite, the set W(x) = {W ∈ W | W ∩ F −1 (x) ̸= ∅} is ﬁnite. Let ε(x) = 21 min{δ(W ) | W ∈ W(x)}. Since the map F is perfect, so is its inverse F −1 : X → exp(Z). By Proposition 3.2.4, F −1 is upper semicontinuous. Hence there is a neighborhood V (x) ⊂ B(x, ε(x)) of x such that F −1 (V (x)) ⊂ ∪W(x). Let us show that the cover V = {V (x)}x∈X is as required, i.e. {St (F(z), V)}z∈Z ≺ U. Fix any z ∈ Z and consider the set St (F(z), V). Let x ∈ X be any point with V (x) ∩ F (z) ̸= ∅. Then z ∈ F −1 (V (x)) ⊂ ∪W(x). Since V (x) ⊂ B(x, ε(x)), where ε(x) = 12 min{δ(W ) | W ∈ W(x)}, we have V (x) ⊂ B(F(z), 2ε(x)) ⊂ B(F(z), δ(W )), where W ∈ W(x) is any set with z ∈ W . Hence St (F(z), V) ⊂ B(F(z), δ(W )) ⊂ B(F(W ), δ(W )) ⊂ U for some U ∈ U. Now we introduce the conception of a (U, F)-map, which generalizes the conception of a U-map. Let F : Z → exp(X) be a multivalued map. Let Y be a space and U an open cover of Y . A map f : Y → X is deﬁned to be a (U, F)-map, if there is an open cover V ∈ cov(X) such that {f −1 (St (F(z), V))}z∈Z ≺ U. A map f : Y → X is deﬁned to be F-injective, if for every z ∈ Z the preimage f −1 (F(z)) contains at most one point. Similarly as in the case of U-maps, the following result holds. 3.2.6. Proposition. The set of all (U, F)-maps is open in C(Y, X). Proof. Let f : Y → X be a (U, F)-map, i.e. there is a cover V ∈ cov(X) such that (1)

{f −1 (St (F(z), V))}z∈Z ≺ U.

Choose any cover W ∈ cov(X) with St W ≺ V. We claim that a map g : Y → X is a (U, F)-map, whenever (g, f ) ≺ W. Indeed, for every

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x ∈ X and every y ∈ g −1 (x) (f, g) ≺ W implies f (y) ∈ St (x, W). Then y ∈ f −1 (St (x, W)) and, consequently, g −1 (St (F(z), W)) ⊂ f −1 (St (St (F(z), W), W)) ⊂ ⊂ f −1 (St (F(z), St W)) ⊂ f −1 (St (F(z), V)). This and (1) imply {g −1 (St (F(z), W))}z∈Z ≺ U, i.e. g is a (U, F)-map.

It is well known that a closed map f : Y → X is a U-map if and only if {f −1 (x)}x∈X ≺ U , see Ex. 4 to §3.2.A. The following Proposition is a “multivalued” counterpart of this statement. 3.2.7. Proposition. Let F : Z → exp(X) be a perfect multivalued map. A closed map f : Y → X is a (U, F)-map, where U ∈ cov(Y ), if and only if {f −1 (F(z))}z∈Z ≺ U. Proof. The “only if” part is trivial. Assume that f : Y → X is a closed map with {f −1 (F(z))}z∈Z ≺ U . For every z ∈ Z ﬁx U (z) ∈ U with f −1 (F(z)) ⊂ U (z). Since the map f is closed, there is an open neighborhood V (F(z)) ⊂ X of F(z) such that f −1 (V (F(z))) ⊂ U (z). Hence, {f −1 (V (F(z)))}z∈Z ≺ U. By Lemma 3.2.5, there is a cover V ∈ cov(X) such that {St (F(z), V)}z∈Z ≺ {V (F(z))}z∈Z . Then {f −1 (St (F(z), V))}z∈Z ≺ {f −1 (V (F(z)))}z∈Z ≺ U, i.e. f is a (U, F)-map. Exercises to §3.2.B. 1. Show that a multivalued map F : Z → exp(X) is upper semicontinuous if and only if for every z ∈ Z and a neighborhood U of F(z) there is a neighborhood V ⊂ Z of z such that F (V ) ⊂ U . 2. Suppose F : X → exp(X) is the multivalued map deﬁned by F (x) = {x}, x ∈ X, and let U ∈ cov(X) be a cover. Show that a map f : X → Y is a (U , F)-map (resp. f is F-injective) if and only if f is a U -map (resp. f is injective).

C. Some facts from Lipschitz geometry of the Hilbert cube. In this subsection and till the end of this section, it is convenient to change the deﬁnitions of the main model spaces Q, s, Qf , and σ (without changing their topological type). We set Q = [0, 1]ω , s = [0, 1)ω , Qf = {(gi ) ∈ Q | qi = 0 for almost all i} ⊂ Q, and σ = Qf ∩ s. The Hilbert cube Q is endowed with the metric d((qi ), (qi′ )) = sup 2−i |qi − qi′ |. i∈N

We will use the fact that the Hilbert cube admits coordinatewise multiplication by reals t ∈ [0, 1].

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For metric spaces (X, d), (Y, ρ) and a real number α > 0 we let Lipα (X, Y ) denote the space of all α-Lipschitz maps X → Y , endowed with the compact open topology. Recall that a map f : X → Y is called α-Lipschitz if ρ(f (x), f (x′ )) ≤ αd(x, x′ ) for every x, x′ ∈ X. If X and Y are compact spaces then Lipα (X, Y ) is a metrizable compactum, according to Ascoli Theorem. 3.2.8. Lemma. There is a homotopy H : Q × [0, 1] → Q such that (1) H(q, 0) = q for every q ∈ Q; (2) H(Q × (0, 1]) ⊂ Qf \s; (3) d(H(q, t), H(q ′ , t′ )) ≤ d(q, q ′ ) + |t − t′ | for every (q, t), (q ′ , t′ ) ∈ Q × [0, 1]. Proof. We deﬁne the homotopy H : Q × [0, 1] → Q as follows. Fix q = −n (qi )∞ ≤ t ≤ 2−(n−1) , i=1 ∈ Q and t ∈ [0, 1]. If t = 0 set H(q, t) = q. If 2 ∞ n ∈ N, let H(q, t) = (H(q, t)i )i=1 , where qi , n n (2 t − 1) + (2 − 2 t)qn+1 , H(q, t)i = 1, 2 − 2n t, 0,

1 ≤ i ≤ n; i = n + 1; i = n + 2; i=n+3 i < n + 3.

By routine veriﬁcation, one can show that the homotopy H satisﬁes conditions (1)—(3). 3.2.9. Lemma. For every compactum A ⊂ s there exists a homotopy HA : Q × [0, 1] → Q such that (1) HA (q, 0) = q for q ∈ Q; (2) HA (Q × (0, 1]) ⊂ σ\A; (3) d(HA (q, t), HA (q ′ , t′ )) ≤ d(q, q ′ ) + |t − t′ | for every (q, t), (q ′ , t′ ) ∈ Q × [0, 1]. Proof. Let H : Q × [0, 1] → Q be a homotopy satisfying conditions ∏∞(1)—(3) of Lemma 3.2.8. Find a sequence (ai )∞ i=1 ⊂ (0, 1) such that A ⊂ i=1 [0, ai ), and deﬁne the homotopy HA : Q × [0, 1] → Q by HA (q, t) = (HA (q, t)i )∞ i=1 , where HA (q, t)i = (tai + 1 − t)H(q, t/2)i for (q, t) ∈ Q × [0, 1]. The reader can readily check that the homotopy HA satisﬁes conditions (1)–(3). 3.2.10. Lemma. Let (K, ρ), ρ ≤ 1, be a metric space, B ⊂ K be a closed subset and f ∈ Lipα (K, Q). For every ε > 0 and a compactum A ⊂ s there is a map f¯ ∈ Lipα+ε (K, Q) such that d(f¯, f ) ≤ ε, f¯|B = f |B and f¯(K\B) ⊂ σ\A.

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Proof. Let HA : Q × [0, 1] → Q be a homotopy satisfying conditions (1)–(3) of Lemma 3.2.9. Deﬁne the map f¯ : K → Q by f¯(x) = HA (f (x), ε · ρ(x, B)) and notice that f¯ is as required. D. Characterizing C-universality for “nice” classes C. The main result of this subsection is 3.2.11. Theorem. Let C be a 2ω -stable weakly A1 -additive M1 -hereditary class of spaces. A pair (M, X) is (M0 ∩C, C)-universal if and only if (M, X) is (M0 ∩ C, C)-preuniversal. Proof. The “only if” part is trivial. Suppose the pair (M, X) is (M0 ∩C, C)preuniversal. Fix a pair (K, C) ∈ (M0 ∩ C, C). We should construct a closed embedding e : K → M such that e−1 (X) = C. Let ρ ≤ 1 be any metric on K. The space Lip1 (K, Q), being a metrizable compactum, is a continuous image of the Cantor cube. Let ξ : 2ω → Lip1 (K, Q) be a surjective map. Consider the map µ : K × 2ω → K × Q deﬁned by µ(x, a) = (x, ξ(a)(x)), (x, a) ∈ K × 2ω . Notice that µ−1 (C × Q) = C × 2ω . Since the class C is 2ω -stable M1 -hereditary and weakly A1 -additive, we have µ−1 (K × σ ∪ C × s) = µ−1 (K × σ) ∪ (µ−1 (C × Q) ∩ µ−1 (K × s)) = µ−1 (K × σ) ∪ (C × 2ω ∩ µ−1 (K × s)) ∈ C, and consequently, (K × 2ω , µ−1 (K × σ ∪ C × s)) ∈ (M0 ∩ C, C). Now we use (M0 ∩ C, C)-preuniversality of the pair (M, X) to ﬁnd a map g : K × 2ω → M such that (1)

g −1 (X) = µ−1 (K × σ ∪ C × s).

Deﬁne a perfect multivalued map F : M → exp(K × Q) letting F(x) = µ ◦ g −1 (x), x ∈ M . It follows from (1) that F ◦ F −1 (K × σ) ⊂ K × s. Now our strategy is to ﬁnd a map f ∈ Lip1 (K, Q) with f (K) ⊂ s\σ such that the map (idK , f ) : K → K × Q is F-injective. Assuming for a moment that such a map f is constructed, ﬁnd a point a ∈ 2ω with ξ(a) = f , and deﬁne a map e : K → M letting e(x) = g(x, a), x ∈ K. We claim that e is the required closed embedding with e−1 (X) = C. Indeed, consider the embedding i : K → K × 2ω acting as i(x) = (x, a), x ∈ K, and notice that e = g ◦ i and µ ◦ i = (idK , f ). Since f (K) ⊂ s\σ, we get (idK , f )−1 (K × σ ∪ C × s) = C, and consequently, e−1 (X) = (g ◦ i)−1 (X) = i−1 (g −1 (X)) = i−1 (µ−1 (K × σ ∪ C × s)) = (µ ◦ i)−1 (K × σ ∪ C × s) = (idK × f )−1 (K × σ ∪ C × s) = C. Now let us verify that the map e is injective. Let x ∈ M be any point. Then e−1 (x) = i−1 ◦ g −1 (x) ⊂ i−1 ◦ µ−1 ◦ µ ◦ g −1 (x) = (µ ◦ i)−1 ◦ F (x) = (idK , f )−1 (F(x)). Since the map (idK , f ) is F-injective, |e−1 (x)| ≤ |(idK × f )−1 (F(x))| ≤ 1. Therefore, e : K → M is an injective map of the compactum K with e−1 (X) = C, i.e. the pair (M, X) is (M0 ∩ C, C)-universal.

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Constructing the map f uses the fact that Lip1 (K, Q) is a Baire space. Let {Un }n∈N ⊂ cov(K) be a sequence of covers such that mesh Un < n1 for ∪ all n. Write σ = n∈N An , where each An is compact, and consider the sets Gn = {f ∈ Lip1 (K, Q) | f (K) ∩ An = ∅}, Hn = {f ∈ Lip1 (K, Q) | (idK , f ) : K → K × Q is a (Un , F)-map}, E = {f ∈ Lip1 (K, Q) | f (K) ⊂ s}. It is easily seen that E is a dense Gδ -set in Lip1 (K, Q). Below we will show that the sets Hn , Gn are open and dense in Lip1 (K, Q). Then, using the fact that Lip1 (K, Q), being ∩ a metrizable compactum, is a Baire space, we get the intersection E ∩ n∈N Gn ∩ Hn is dense in Lip1 (K, Q), and hence contains a function f . Evidently, f is as required, i.e., f (K) ⊂ s\σ and (idK , f ) is F-injective. The sets Gn ’s. Evidently, each set Gn is open in Lip1 (K, Q). To show that it is dense, ﬁx any f ∈ Lip1 (K, Q), and an ε > 0. Then the map (1 − 2ε )f : K → s is 2ε -near to f and is (1 − 2ε )-Lipschitz. Now applying Lemma 3.2.10, ﬁnd a map f ′ ∈ Lip1 (K, Q), 2ε -close to (1 − 2ε )f and such that f ′ (K) ∩ An = ∅. Then f ′ ∈ Gn is ε-close to f , i.e. Gn is open and dense in Lip1 (K, Q). The sets Hn ’s. To show that each Hn is open and dense in Lip1 (K, Q), we will prove that for every U ∈ cov(K) the set H(U) = {f ∈ Lip1 (K, Q) | (idK , f ) : K → K × Q is a (U, F)-map} is open and dense in Lip1 (K, Q). By Proposition 3.2.6, the set of all (U, F)-maps is open in C(K, K × Q). Since the map Lip1 (K, Q) → C(K, K × Q) sending an f onto (idK , f ) is continuous, the set H(U) is open in Lip1 (K, Q). Let us show that H(U) ⊂ Lip1 (K, Q) is dense. Fix ε > 0 and a map f ∈ Lip1 (K, Q). We have to ﬁnd a map f¯ ∈ H(U) with d(f, f¯) ≤ ε. Denote by prK : K × Q → K and prQ : K × Q → Q the natural projections. Let {A1 , . . . , Am } ≺ U be a ﬁnite closed cover of the compactum K, inscribed into the cover U. For 1 ≤ n ≤ m let Kn = A1 ∪ · · · ∪ An . The required map f¯ ∈ Lip1 (K, Q) will be constructed by ﬁnite induction. Let f−1 = (1 − ε/2) · f : K → Q. Evidently, d(f−1 , f ) ≤ ε/2 and f ∈ ε Lip1−ε/2 (K, Q). Let δ = 2(m+1) . By Lemma 3.2.10, there is a map f0 ∈ Lip1−ε/2+δ (K, Q) such that d(f0 , f−1 ) ≤ δ and f0 (K) ⊂ σ. Using the fact F ◦ F −1 (K × σ) ⊂ K × s and Lemma 3.2.10, construct inductively a sequence of maps {fn : K → σ}m n=1 such that fn |Kn = fn−1 |Kn , d(fn , fn−1 ) ≤ δ, fn ∈ Lip1−ε/2+(n+1)δ (K, Q), fn (K\Kn ) ∩ prQ ◦ F ◦ F −1 (Kn × fn−1 (Kn )) = ∅.

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Then the map f¯ = fm has the following properties: d(f¯, f ) ≤ d(fm , f−1 )+ d(f−1 , f ) ≤ (m + 1)δ + ε/2 = ε, f¯ ∈ Lip1 (K, Q) and (2)

f¯(K\Kn ) ∩ prQ ◦ F ◦ F −1 (Kn × f¯(Kn )) = ∅ for every n ∈ N.

Let us show that idK × f¯ : K → K × Q is a (U, F)-map. According to 3.2.7, for this it suﬃces to show that {(idK × f¯)−1 (F(z))}z∈Z ≺ U. Fix any z ∈ Z, and let D = (idK × f¯)−1 (F(z)). Let n = min{i ∈ N | D ∩ Ai ̸= ∅}. We claim that D ⊂ An . Fix any point x ∈ D ∩ An . Then (x, f¯(x)) ∈ F (z), and, hence, z ∈ F −1 (x, f¯(x)) ⊂ F −1 (Kn × f¯(Kn )). Notice that D = (idK × f¯)−1 (F(z)) implies D ⊂ f¯−1 ◦prQ ◦F (z) ⊂ f¯−1 ◦prQ ◦F ◦F −1 (Kn × f¯(Kn )). This and (2) yield D ⊂ Kn = A1 ∪ · · · ∪ An . Since D ∩ (A1 ∪ · · · ∪ An−1 ) = ∅, we have D ⊂ An . Therefore, {(idK × f¯)−1 (F(z))}z∈Z ≺ U , i.e. idK × f¯ is a (U, F)-map. Now we are going to formulate a “space” counterpart of Theorem 3.2.11. For this we need to introduce a notion of a D-map. A map f : X → Y is deﬁned to be a D-map, where D is a class of spaces, if there are a space D ∈ D and a closed embedding e : X → Y × D such that f = prY ◦ e (here prY : Y × D → Y denotes the natural projection). Notice that a map f : X → Y is an M0 -map if and only if f is perfect, i.e. f −1 (K) ∈ M0 for every compact subset K ⊂ Y ; Similarly, a map f : X → Y between Borel spaces is an M1 -map if and only if f −1 (K) ∈ M1 for every compact K ⊂ Y , see Ex.3. 3.2.12. Theorem. Let C be a 2ω -stable M1 -hereditary weakly A1 -additive compactiﬁcation-admitting class of spaces. For a space X the following conditions are equivalent: (1) the space X is C-universal; (2) the product X × s is C-universal; (3) there is an M1 -map f : Y → X of a C-universal space Y into X; (4) for every C ∈ C there is an M1 -map f : C → X. Proof. The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 2) are trivial (for the proof of the last one use M1 -universality of s). 2) ⇒ 1). Suppose X × s is C-universal. Let M be any completion of X. Then M × s is a Polish space containing the C-universal space X × s. By Theorem 3.1.1, the pair (M × s, X × s) is (M0 ∩ C, C)-universal. Then the pair (M, X) is (M0 ∩ C, C)-preuniversal, and by Theorem 3.2.11, this pair is (M0 ∩ C, C)-universal. Since the class C admits compactiﬁcations, this implies the space X is C-universal. Among important for applications classes that do not satisfy the assumptions of Theorem 3.2.12 there are the Borel classes M0 , A1 , and M1 . Fortunately, for the class M1 , a similar result to Theorem 3.2.12 still holds.

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3.2.13. Theorem. For a space X the following conditions are equivalent: (1) the space X is M1 -universal; (2) the product X × Σ is M1 -universal; (3) there is an A1 -map f : Y → X of an M1 -universal space Y into X; (4) for every C ∈ M1 there is an A1 -map f : C → X; (5) there is a perfect map p : s → X. Proof. The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 2) are trivial (for the proof of the last one use A1 -universality of Σ). ∪ 2) ⇒ 5). Let e : s → X × Σ be a closed embedding. Write Σ = n∈N Qn , where each Qn is compact. By Baire Theorem, one of e−1 (X × Qn ) has a non-empty interior in s, and hence contains a closed topological copy T of s. Then the restriction prX ◦ e|T : T → X is a perfect map, and thus 5) holds. 5) ⇒ 1). Let p : s → X is a perfect map. To show that the space X is M1 -universal, ﬁx a Polish space C. We should construct a closed embedding e : C → X. For this we will construct a closed embedding f : C → s such that the composition p ◦ f : C → X is injective. Then e = p ◦ f : C → X, being a perfect injective map, is just a required closed embedding. Fix a complete metric d on C and let {Un }n∈N ⊂ cov(s) be a sequence of covers with mesh Un < n1 for all n. Consider the perfect multivalued map F = p−1 : X → exp(s). Consider for every n ∈ N the set Hn = {f ∈ C(C, s) | f is a (Un , F)-map}. Below (in Lemma 3.2.14) we will prove that each Hn is open and dense in C(C, s). By Proposition 3.2.3, the set E of all closed ∩ embeddings is a dense Gδ -set in C(C, s). Then by 3.2.1, the set E ∩ n∈N Hn is dense in C(C, s), and thus contains a function f . Evidently, f is an F-injective closed embedding. Clearly, F-injectivity of f implies injectivity of the map p ◦ f. 3.2.14. Lemma. Let M be an ANR with SDAP, and F : Z → exp(M ) a perfect multivalued map. For every space X and every U ∈ cov(X) the set H(U) = {f ∈ C(X, M ) | f is a (U, F)-map} is open and dense in C(X, M ). Proof. By 3.2.6, the set H(U) is open in C(X, M ). To show that H(U) is dense in C(X, M ), ﬁx a map f ∈ C(X, M ) and a cover V ∈ cov(M ). Let V ′ ∈ cov(M ) be a cover with St V ′ ≺ V. By 1.1.9, there are a locally compact space K, a U-map p : X → K, and a map q : K → M such that (q ◦ p, f ) ≺ V ′ . According to 1.3.4, we can assume the map q to be perfect.

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Let W ∈ cov(K) be such that p−1 W ≺ U. By Ex.7 to §1.1.A, there is a cover V0 ∈ cov(M ), V0 ≺ V ′ , such that a map q¯ : K → M is perfect whenever (¯ q , q) ≺ V0 . Let {Vn }n∈N ⊂ cov(M ) be a sequence of covers such that St Vn ≺ Vn−1 and mesh Vn < 2−n for all n. Let {Bn }n∈N ≺ W be a cover of K by compact sets. For every n ∈ N let Kn = B1 ∪ · · · ∪ Bn . Inductively, using homotopical negligibility of compact sets in M , construct a sequence {qn : K → M }∞ n=1 such that (1)

qn |Kn = qn−1 |Kn , (qn , qn−1 ) ≺ Vn , qn (K\Kn ) ∩ F ◦ F −1 ◦ qn−1 (Kn ) = ∅.

Consider the limit map q¯ = limn→∞ qn : K → M . It is easily seen that q¯ is continuous, (¯ q , q0 ) ≺ V0 , and (2)

q¯(K\Kn ) ∩ F ◦ F −1 ◦ q¯(Kn ) = ∅ for every n ∈ N.

By the choice of the cover V0 , the map q¯ is perfect. Similarly as in the proof of Theorem 3.2.11, show that {¯ q −1 (F(z))}z∈Z ≺ {Bn }n∈N ≺ W. By Proposition 3.2.7, q¯ is a (W, F)-map. Since p−1 W ≺ U, the composition q¯ ◦ p : X → M is a (U, F)-map with (¯ q ◦ p, f ) ≺ St V ′ ≺ V. Theorem 3.2.13 has a ﬁnite-dimensional analog. Recall that M1 [n] denotes the class of Polish spaces with dim ≤ n. 3.2.15. Theorem. Let n ≥ 0. For a space X the following conditions are equivalent: (1) the space X is M1 [n]-universal; (2) the product X × Σ is M1 [n]-universal; (3) there is an A1 -map f : Y → X of an M1 [n]-universal space Y into X; (4) for every C ∈ M1 [n] there is an A1 -map f : C → X; (5) for every C ∈ M1 [n] there is a perfect map p : C → X. Proof. The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 2) are trivial. 2) ⇒ 5). Fix any C ∈ M1 [n]. Assuming that the product X × Σ in M1 [n]universal, ﬁnd a closed embedding ∪ e : Nn → X × Σ of the n-dimensional N¨obeling space Nn . Write Σ = k∈N Qk , where each Qk is compact. By Baire Theorem, there is a k such that the set e−1 (X × Qk ) has non-empty interior in Nn . By 2.3.5, there is an n-invertible map µ : M → Q of an n-dimensional compactum M onto the Hilbert cube. Obviously, µ−1 (s) ∈ M1 [n]. Since the N¨obeling space Nn is everywhere M1 [n]universal (see Ex.4 to §2.2), there exists a closed embedding j : µ−1 (s) →

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e−1 (X × Qk ) ⊂ Nn . Using M1 -universality of s, ﬁnd a closed embedding α : C → s, and by n-invertibility of µ, ﬁnd a map β : C → µ−1 (s) such that α = µ ◦ β. It is easily seen that β is a closed embedding. Then the composition p = prX ◦ e ◦ j ◦ β : C → X is a perfect map (here prX : X × Σ → X stands for the natural projection). 5) ⇒ 1). Fix C ∈ M1 [n]. Let p : µ−1 (s) → X be a perfect map. Deﬁne a perfect multivalued map F : X → exp(s) by the formula F(x) = µ ◦ p−1 (x), x ∈ X. Similarly as in Theorem 3.2.13, ﬁnd an F-injective closed embedding f : C → s. Using n-invertibility of µ, ﬁnd a map h : C → µ−1 (s) such that f = µ ◦ h. One can easily see that h is a closed embedding. Consider ﬁnally the map e = p ◦ h : C → X and remark that this map is perfect as a composition of two perfect maps. Moreover, for every x ∈ X we have e−1 (x) = h−1 ◦ p−1 (x) ⊂ h−1 ◦ µ−1 ◦ µ ◦ p−1 (x) = (µ ◦ h)−1 ◦ (µ ◦ p−1 )(x) = f −1 ◦ F(x) and, consequently, |e−1 (x)| ≤ 1. Therefore, e being a perfect injective map, is a closed embedding. Exercises and Problems to §3.2.D. 1. Let C, D be two classes of spaces, and let D be a D-universal space. Show that for a space X the following conditions are equivalent: (a) the product X × D is C-universal; (b) there is a D-map f : Y → X of a C-universal space Y ; (c) for every C ∈ C there is a D-map f : C → X. 2. Show that a map f is perfect if and only if f is an M0 -map. 3. Let C be one of Borel classes Mα , Aα . Show that for a map f : X → Y the following conditions are equivalent: (a) f is a C-map; (b) there is a C-map g : Z → Y and an embedding e : X → Z such that g ◦ e = f and e(X) ∈ C(Z); (c) for every C-map g : Z → Y and an embedding e : X → Z with g ◦ e = f we have e(X) ∈ C(Z). Moreover, if the spaces X, Y are Borel (in the case C = M0 no additional assumptions are needed, if C = M1 we can assume X to be coanalytic and Y analytic, if C = A1 , we can suppose X to be analytic and Y coanalytic), then the conditions (a)—(c) are equivalent to (d) f −1 (K) ∈ C(X) for every compact subset K ⊂ Y . Hint: Use Hurewicz Separation Theorem 28.18 and Ex.22.13 from A.Kechris [1995]. A map f : X → Y is called Borel of the class 1 if for every closed subset F ⊂ Y its preimage f −1 (F ) is a Gδ -subset in X. 4. Suppose f is a bijective map whose inverse f −1 is Borel of the class 1. Show that f is an M1 -map. 5. Suppose C is a 2ω -stable weakly A1 -additive Gδ -hereditary compactiﬁcation-admitting class of spaces such that for every C ∈ C there is an everywhere C-universal Baire ˜ ∈ σC; Show that for a space X the following conditions are equivalent: (a) space C X is C-universal; (b) X × s × σ is C-universal; (c) there is an M21 -map f : Y → X of a C-universal space Y ; (d) for every C ∈ C there is an M21 -map f : C → X. 6. Suppose n ∈ N and C is a 2ω -stable weakly P2n−1 -additive Gδ -hereditary compactiﬁcation-admitting class of spaces. Show that for a space X the following conditions are equivalent: (a) X is C-universal; (b) X × Π2n is C-universal; (c) there is a P2n−1 map f : Y → X of a C-universal space Y ; (d) for every C ∈ C there is a P2n−1 -map f : C → X. Hint: Use Ex.2 to §3.1.B.

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A map f : X → Y is called C-invertible, where C is a space, if for every map α : C → Y there is a map β : C → X with f ◦ β = α. 7. Suppose C ⊂ M1 is a class of Polish spaces such that for every C ∈ C there exists ˜ → s, where C ˜ ∈ C. Show that for a space X the a C-invertible perfect map f : C following conditions are equivalent: (a) X is C-universal; (b) X × Q is C-universal; (c) there is an M0 -map f : Y → X of a C-universal space Y ; (d) for every C ∈ C there is an M0 -map f : C → X. 8. Suppose C ⊂ M1 is a class of Polish spaces such that for every C ∈ C there are an ˜1 ∈ C and a C-invertible perfect map f : C ˜2 → s, everywhere C-universal space C ˜2 ∈ C. Show that for a space X the following conditions are equivalent: (a) where C X is C-universal; (b) X × Σ is C-universal; (c) there is an A1 -map f : Y → X of a C-universal space Y ; (d) for every C ∈ C there is an A1 -map f : C → X. 9. (Open Problem) Is Theorem 3.2.15 valid for transﬁnite dimensions? 10. (Open Problem) Suppose C ∈ B\{A1 } is a Borel class and C ◦ its dual (see Ex.2 to §3.1.D). Is a space X C-universal if X × C is C-universal for some C ∈ C ◦ ? 11. A non-compactness measure. For a space X let nc(X) the minimal number n (ﬁnite or inﬁnite) for which there exists a perfect map of X onto an n-dimensional space. Prove the following properties of the function nc: a) nc(X) = 0 for every compactum; b) nc(X) ≤ 1 for every locally compact space X; c) nc(X) ≤ nc(Y ) for every closed subspace X ⊂ Y ; d) for every perfect surjective map p : X → Y we have nc(X) = nc(Y ); e) nc(X × Y ) ≤ nc(X) + nc(Y ) for every spaces X, Y ; f) nc(A ∪ B) ≤ nc(A) + nc(B) + 2 for every closed subspaces A, B ⊂ X; g) nc(Nn ) = n for the n-dimensional N¨ obeling space Nn ; h) nc(s) = ∞.

E. Characterizing strong C-universality for “nice” classes C. In this subsection we prove “strong” versions of results from the previous subsection. Given classes K ⊂ M0 and C, we deﬁne a pair (M, X) to be strongly (K, C)-preuniversal, if for every pair (K, C) ∈ (K, C), every closed subset B ⊂ K, every cover U ∈ cov(M ), and every map f : K → M such that (f |B)−1 (X) = B ∩ C there is a map f¯ : K → M with the properties (f¯, f ) ≺ U, f¯|B = f |B, and f¯−1 (X) = C. 3.2.16. Theorem. Let C be a 2ω -stable M1 -hereditary weakly A1 -additive class of spaces. For a pair (M, X), where M is an ANR and X is a homotopy dense subset in M , the following conditions are equivalent: (1) the pair (M, X) is strongly (M0 ∩ C, C)-universal; (2) the pair (M, X) is strongly (M0 ∩ C, C)-preuniversal; (3) for a cover U ∈ cov(M ), a pair (K, C) ∈ (M0 ∩ C, C), a closed subspace B ⊂ K and a map f : K → M such that f (B) is a Z˜ such that (f¯, f ) ≺ U , f¯|B = f |B, set, there is a map f¯ : K → X −1 ¯ ¯ f (K\B) ∩ f (B) = ∅, and f (X)\B = C\B.

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Proof. The implications 1) ⇒ 2) ⇒ 3) are easy and can be proved by the methods elaborated in the previous sections. 3) ⇒ 1) Suppose the pair (M, X) satisﬁes condition 3). To prove strong (M0 ∩ C, C)-universality of (M, X), ﬁx a cover U ∈ cov(M ), a pair (K, C) ∈ (M0 ∩ C, C), a closed subset B ⊂ K and a map f : K → M such that f |B : B → M is a Z-embedding with (f |B)−1 (X) = B ∩ C. Let ρ ≤ 1 be any metric on K. Since Lip1 (K, Q) is a metrizable compactum, there is a surjection ξ : 2ω → Lip1 (K, Q). Consider the map µ : K ×2ω → K ×Q deﬁned by µ(x, a) = (x, ξ(a)(x)), (x, a) ∈ K ×2ω . Analogously as in the proof of Theorem 3.2.12, show that (K×2ω , µ−1 (K×σ∪C× s)) ∈ (M0 ∩C, C). By the condition (3), there is a map p : K ×2ω → M such that (p, f ◦ prK ) ≺ U, p|B × 2ω = f ◦ prK |B × 2ω , p((K\B) × 2ω ) ∩ f (B) = ∅, and p−1 (X\f (B)) = µ−1 ((K × σ ∪ C × s)\(B × 2ω )). Deﬁne a perfect multivalued map F : M → exp(K × Q) by the formula F(z) = µ ◦ p−1 (x), x ∈ M . Notice that F ◦ F −1 ((K\B) × σ) ⊂ K × s and F ◦ F −1 ({b} × Q) ⊂ {b} × Q for every b ∈ B. ∪ Let A be a countable dense set in C(Q, X\f (B)) and let A′ = α∈A α(Q) and A = prQ ◦ µ ◦ p−1 (A′ ), where prQ : K × Q → Q is the projection. It is easily seen that A is a σ-compactum in s. Similarly as in Theorem 3.2.12, our strategy now consists in ﬁnding a map g ∈ Lip1 (K, Q) such that g(K) ⊂ s\(σ ∪ A) and the map (idK , g) : K → K × Q is F-injective. Assuming for a moment that such a map g is constructed, pick up any a ∈ ξ −1 (g) and deﬁne a map f¯ : K → M letting f¯(x) = p(x, a), x ∈ K. We claim that f¯ is the required Z-embedding such that f¯|B = f |B, (f¯, f ) ≺ U and f¯−1 (X) = C. The properties f¯|B = f |B and (f¯, f ) ≺ U easily follow from the corresponding properties of the map p and the deﬁnition of f¯. Equality f¯−1 (X)\B = C\B and injectivity of f¯ can be veriﬁed analogously as in the proof of Theorem 3.2.11. To see that f¯(K) is a Z-set in M , notice that f¯(K) ∩ A′ = ∅. Since the set A is dense in C(Q, X\f (B)) and the set ∪ X\f (B) is homotopy dense in M , A is dense in C(Q, M ). Since f¯(K) ∩ α∈A α(Q) = ∅, this yields that f¯(K) is a Z-set in M . Constructing the map g is similar to the construction of the map f in Theorem 3.2.12. Let (Un )n∈N ⊂ ∪cov(K) be a sequence of covers with ∞ mesh Un < n1 , n ∈ N. Write σ ∪ A = n=1 An , where An are compacta, and

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consider the sets Gn = {f ∈ Lip1 (K, Q) | f (K) ∩ An = ∅}, Hn = {f ∈ Lip1 (K, Q) | (idK , f ) : K → K × Q is a (Un , F)-map}, E = {f ∈ Lip1 (K, Q) | f (K) ⊂ s}. Like as in Theorem 3.2.11, we conclude that E is a dense Gδ -set in Lip1 (K, Q) and the sets Gn are open and dense in Lip1 (K, Q). It follows from Lemma 3.2.17 (see below) that the same is true for the sets H∩ n . Then using the fact that Lip1 (K, Q) is a Baire space, pick up any g ∈ E ∩ n∈N Gn ∩ Hn . Evidently that g(K) ⊂ s\(A ∪ σ) and the map (idK , g) is F-injective. 3.2.17. Lemma. Let (K, ρ) be a metric compactum with ρ ≤ 1, B ⊂ K be a closed subset in K and F : Z → exp(K × Q) be a perfect multivalued map such that F ◦ F −1 ((K\B) × σ) ⊂ K × s and F ◦ F −1 ({b} × Q) ⊂ {b} × Q for every b ∈ B. Then for every cover U ∈ cov(K) the set H(U) = {f ∈ Lip1 (K, Q) | (idK , f ) : K → K × Q is a (U, F)-map} is open and dense in Lip1 (K, Q). Proof. By 3.2.6, the set of all (U, F)-maps is open in C(K, K ×Q). Since the map Lip1 (K, Q) → C(K, K × Q) sending an f onto (idK , f ) is continuous, the set H(U) is open in Lip1 (K, Q). Let us show that H(U) ⊂ Lip1 (K, Q) is dense. Fix ε > 0 and a map f ∈ Lip1 (K, Q). We have to ﬁnd a map f¯ ∈ H(U) with d(f, f¯) ≤ ε. Denote by prK : K × Q → K and prQ : K × Q → Q the natural projections. Notice that the multivalued map G : K → exp K deﬁned by G(x) = prK ◦ F ◦ F −1 ({x} × Q), x ∈ K, is perfect, and hence is uppersemicontinuous. Since for every x ∈ B we have G(x) ⊂ {x}, there is a neighborhood Wx ⊂ K of x such that (1)

G(Wx ) ⊂ Ux for some Ux ∈ U.

∪ Let W = x∈B Wx . Since F is perfect, the set A = prK ◦F◦F −1 ((K\W )× Q) is closed in K. Moreover, since F ◦ F −1 (B × Q) ⊂ B × Q we see that A∩B = ∅. Let {A1 , . . . , Am } ≺ U be a ﬁnite closed cover of the compactum A, inscribed into the cover U. For 1 ≤ n ≤ m let Kn = A1 ∪ · · · ∪ An . The required map f¯ ∈ Lip1 (K, Q) will be constructed by ﬁnite induction. Let f−1 = (1 − ε/2) · f : K → Q. Evidently, d(f−1 , f ) ≤ ε/2 and f ∈ ε Lip1−ε/2 (K, Q). Let δ = 2(m+1) . By Lemma 3.2.10, there is a map f0 ∈ Lip1−ε/2+δ (K, Q) such that d(f0 , f−1 ) ≤ δ and f0 (K) ⊂ σ.

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Using Lemma 3.2.10, construct inductively a sequence of maps {fn : K → σ}m n=1 such that fn |Kn = fn−1 |Kn , d(fn , fn−1 ) ≤ δ, fn ∈ Lip1−ε/2+(n+1)δ (K, Q), fn (K\Kn ) ∩ prQ ◦ F ◦ F −1 (Kn × fn−1 (Kn )) = ∅. Then the map f¯ = fm has the following properties: d(f¯, f ) ≤ d(fm , f−1 )+ d(f−1 , f ) ≤ (m + 1)δ + ε/2 = ε, f¯ ∈ Lip1 (K, Q) and (2)

f¯(K\Kn ) ∩ prQ ◦ F ◦ F −1 (Kn × f¯(Kn )) = ∅ for every n ∈ N.

Let us show that (idK , f¯) : K → K × Q is a (U, F)-map. According to Proposition 3.2.7, for this it suﬃces to show that {(idK , f¯)−1 (F(z))}z∈Z ≺ U. Fix any z ∈ Z, and let D = (idK , f¯)−1 (F(z)). We will consider separately two cases: F(z) ⊂ W × Q and F(z) ̸⊂ W × Q. In the ﬁrst case, by (1), D ⊂ prK (F(z)) ⊂ U for some U ∈ U. If F(z) ̸⊂ W × Q then D ⊂ prK ◦ F ◦ F −1 (K\W × Q) = A. Let n = min{i ∈ N | D ∩ Ai ̸= ∅}. Similarly as in Theorem 3.2.11, we can show that D ⊂ An . Then {(idK , f¯)−1 (F(z))}z∈Z ≺ U, i.e. (idK , f¯) is a (U, F)-map. For a class of spaces C, a map f : X → Y is deﬁned to be approximatively C-invertible, if for every map p : C → Y , where C ∈ C, and for every cover U ∈ cov(Y ) there is a map q : C → X with (f ◦q, p) ≺ U. A map f : X → Y is called strongly approximatively C-invertible, if for every open set U ⊂ Y the map f |f −1 (U ) : f −1 (U ) → U is approximatively C-invertible. 3.2.18. Theorem. Let C be a 2ω -stable M1 -hereditary weakly A1 -additive compactiﬁcation-admitting class of spaces. Then for a space X ∈ ANR satisfying SDAP the following conditions are equivalent: (1) X is strongly C-universal; (2) the product X × s is strongly C-universal; (3) there is a strongly approximatively C-invertible M1 -map f : Y → X of a strongly C-universal space Y ; (4) for every open set U ⊂ X, every cover U ∈ cov(U ), and every map f : C → U , where C ∈ C, there is an M1 -map f¯ : C → U such that (f¯, f ) ≺ U. Proof. The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are rather trivial (the ﬁrst implication follows from 1.5.11, and the last one from 1.5.3). 4) ⇒ 1). Suppose an ANR X having SDAP satisﬁes the condition 4). According to Theorem 1.2.4, there is a Polish ANR M containing X as a homotopy dense subspace. We claim that the pair (M, X) satisﬁes the condition (3) of Theorem 3.2.16. Let U ∈ cov(M ), (K, C) ∈ (M0 ∩ C, C), B ⊂ K be a closed subset, and let f : K → M be a map such that f (B)

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is a Z-set in M . Since X\f (B) is homotopy dense in M (see Ex.4 to §1.2), we may assume that f (K\B) ⊂ X\f (B). Let V ∈ cov(M \f (B)) be a cover such that V ≺ U and V ≺ {B(x, d(x, f (B))/2) | x ∈ M \f (B)} (here d is any metric on M ). Denote by prK : K × 2ω → K and pr2ω : K × 2ω → 2ω the natural projections. Let also K ′ = K\B, C ′ = C\B, X ′ = X\f (B) and M ′ = M \f (B). Let A be a countable dense set in 2ω . Notice that the subspace C ′ × 2ω ∪ ′ K × A belongs to the class C. Then, by condition (4), there is an M1 -map g : C ′ × 2ω ∪ K ′ × A → X ′ such that (g, f ◦ prK ) ≺ V.

(1)

By the deﬁnition of an M1 -map, there are a Polish space D and a closed embedding e : C ′ × 2ω ∪ K ′ × A → X ′ × D such that g = prX ′ ◦ e. It follows from Lavrentiev Theorem that the embedding e extends to an embedding e¯ : P → M ′ ×D of some Gδ -subset P ⊂ K ′ ×2ω containing C ′ ×2ω ∪K ′ ×A. Since e is a closed embedding, we have (2)

e¯−1 (X ′ × D) = C ′ × 2ω ∪ K ′ × A

see Ex.10 to §1.1.A. Moreover, because of (1), we can assume that (3)

(prM ′ ◦ e¯, f ◦ prK |P ) ≺ V.

Notice that the complement (K ′ × 2ω )\P as well as its projection S onto 2ω are σ-compact. Since P ⊃ K ′ × A, we have S ⊂ 2ω \A. Noticing that 2ω \A ̸∈ A1 , we obtain that there exists t0 ∈ 2ω \(A ∪ S). This yields K ′ × {t0 } ⊂ P . Deﬁne ﬁnally the map f¯ : K → M by the formula { f¯(k) =

f (k),

if k ∈ B;

prM ◦ e¯(k, t0 ),

if k ∈ K\B.

Because of (3) the map f¯ is continuous and is U-close to f . On the other hand, the equality (2) implies f¯−1 (X)\B = C\B, i.e. the pair (M, X) satisﬁes condition (3) of Theorem 3.2.16. Consequently, it is strongly (M0 ∩ C, C)-universal. Applying Theorem 1.7.9, we conclude that the space X is strongly C-universal. Let us prove ﬁnally “strong” counterparts of Theorems 3.2.13 and 3.2.15. 3.2.19. Theorem. For every 0 ≤ n ≤ ∞ and every ANR-space X possessing the strong Z-set property, the following conditions are equivalent: (1) X is strongly M1 [n]-universal;

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123

(2) the product X × Q is strongly M1 [n]-universal; (3) there is a strongly approximatively M1 [n]-invertible perfect map f : Y → X of a strongly M1 [n]-universal space Y ; (4) for every open set U ⊂ X, every cover U ∈ cov(U ), and every map f : C → U , where C ∈ M1 [n], there is a perfect map f¯ : C → U such that (f¯, f ) ≺ U. Proof. The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are trivial. 4) ⇒ 1). Suppose X satisﬁes the condition 4). Since the class M1 [n] is Nω -stable and open-hereditary, by 3.1.6, to prove the strong M1 [n]universality of X, it is enough for given open set U ⊂ X, a cover U ∈ cov(U ), and a map f : C → U of a space C ∈ M1 [n] to ﬁnd a closed embedding f¯ : C → U , U-close to f . Embed the Polish space C as a closed set into s. Since U is an ANR, there are an open set M ⊂ s containing C and a map f˜ : M → U extending the map f . Now we use the fact of the existence of a perfect n-invertible map µ : C˜ → M , where C˜ ∈ M1 [n] (if n = ∞, we let C˜ = M and µ = id : C˜ → M ; if n < ∞, we set µ = η|C˜ : C˜ → M , where η : K → Q is an n-invertible map of an n-dimensional compactum K onto Q, and C˜ = η −1 (M ), see 2.3.5). Let V ∈ cov(U ) be a cover such that St V ≺ U. By the hypothesis, there is a perfect map p : C˜ → U such that (p, f˜ ◦ µ) ≺ V. Deﬁne a perfect multivalued map F : U → exp(M ) by F(x) = µ◦p−1 (x), x ∈ U . Using Lemma 3.2.14, analogously as in Theorem 3.2.13, ﬁnd an Finjective closed embedding g : C → M such that (g, idC ) ≺ f˜−1 (V). Since the map µ is n-invertible, there is a map g ′ : C → C˜ such that µ ◦ g ′ = g. It is easily seen that the map g ′ is perfect. We claim that f¯ = p ◦ g ′ : C → U is a closed embedding with (f¯, f ) ≺ U. Indeed, (g, idC ) ≺ f˜−1 (V) implies that (f ◦ g, f ) ≺ V. On the other hand, (p, f˜ ◦ µ) ≺ V implies that (f¯, f ◦ g) = (p ◦ g ′ , f˜ ◦ µ ◦ g ′ ) ≺ V. Consequently, (f¯, f ) ≺ St V ≺ U. The map f¯ = p ◦ g ′ : C → U is perfect as a composition of perfect maps. Since |f¯−1 (x)| = |(g ′ )−1 ◦ p−1 (x)| ≤ |(g ′ )−1 ◦ µ−1 ◦ µ ◦ p−1 (x)| = |g −1 ◦ F(x)| ≤ 1, x ∈ U , f¯ is a closed embedding. Exercises and Problems to §3.2.E. 1. Suppose n ∈ N and C is a 2ω -stable P2n−1 -hereditary weakly A1 -additive compacti-

ﬁcation-admitting class of spaces. Show that for a space X ∈ ANR with SDAP the following conditions are equivalent: (a) X is strongly C-universal; (b) the product X × Π2n is strongly C-universal; (c) there is a strongly approximatively C-invertible P2n−1 -map f : Y → X of a strongly C-universal space Y ; (d) for every open set U ⊂ X and every C ∈ C the set of all P2n−1 -maps is dense in C(C, U ).

2. Suppose C ⊂ M1 is an open-hereditary class of Polish spaces such that for every C ∈ C

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˜ → s, where C ˜ ∈ C. Suppose X is an there exists a C-invertible perfect map µ : C ANR whose every Z-set is strong. Show that the following conditions are equivalent: (a) X is strongly C-universal; (b) the product X × Q is strongly C-universal; (c) there is a strongly approximatively C-invertible perfect map f : Y → X of a strongly Cuniversal space Y ; (d) for every open set U ⊂ X and every C ∈ C the set of all perfect maps is dense in C(C, U ). 3. Show that a space X satisﬁes SDAP if and only if there exists an approximatively M0 (ω)-invertible perfect map f : Y → X of a space Y with SDAP. 4. Show that every retraction is strongly approximatively M-invertible. 5. Show that Λ1 (ω) × Q ∼ ̸ Λα and Ωα (ω) × Q ∼ ̸ Ωα . = Λ1 , but for α > 1 Λα (ω) × Q ∼ = =

Notes and Comments to Chapter III. Results of the subsections 3.1.A–3.1.C can be found in T.Banakh, R.Cauty [2000]. All other results belong to T.Banakh.

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Chapter IV Applications I

This chapter contains the ﬁrst group of applications of the theory of absorbing spaces and pairs: to inﬁnite products, to topological groups, and to hyperspaces. §4.1. Products of absolute retracts In this section we investigate the topology of countable products of absolute retracts. Toru´ nczyk’s Characterization Theorems for Q and l2 imply the following statements: a) the countable product of inﬁnite compact AR’s is homeomorphic to Q; b) the countable product of complete noncompact AR’s is homeomorphic to l2 (see Ex.3 to §1.1.C). It is natural to ask what happens if the factors are of higher Borel classes. Note ﬁrst that X ω ∈ Mξ whenever X ∈ Mξ . 4.1.1. Theorem. Let X ∈ M2 be an AR-space which is a countable union of its strong Z-sets. Then X ω is homeomorphic to Ω2 . We need the following auxiliary result. 4.1.2. Lemma. Let X ∈ AR, X ̸= {point}. Then X ω is strongly M0 (X)universal. Proof. We identify X ω with (X ω )ω and hence represent each point in X ω ω ω as (xi )∞ i=1 where xi ∈ X . Let d be a metric on X with the property: ∞ ∞ −k−2 d((xi )i=1 , (yi )i=1 ) ≤ 2 whenever xi = yi for each i ≤ k. Assume that a closed embedding g : C → X ω is given. Let f : C → X ω be a map such that f |D : D → X ω is a Z-embedding and let ε : X ω → (0, 1) be a Lipschitz map. ˜ such that X is There exists an embedding of X into a complete AR X ˜ (see 1.2.4) and hence there exists an embedding of X ω homotopy dense in X ˜ ω such that X ω is homotopy dense in X ˜ ω . Since X ˜ ω is homeomorphic in X to Q or s (see above), by Ex 16 and 17 to §1.4, every Z-set in X ω is a strong Z-set. Hence, by 1.4.7, we can assume that f (C \ D) ∩ f (D) = ∅ and the map f is closed over f (D).

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APPLICATIONS I

For each x ∈ X ω let δ(x) = min{ε(x), d(x, f (D))}. Fix a point ∗ ∈ X ω . There exists a homotopy G : C × [0, 1] → X ω such that G(c, 0) = g(c) and G(c, 1) = (∗, ∗, ∗, . . . ). We write f = (fi )∞ i=1 where fi are coordinate maps. There exist homotopies Hi : C × [0, 1] → X ω such that Hi (c, 0) = g(c) and Hi (c, 1) = fi (c). Deﬁne f ′ : C → X ω by the following manner. For 2−k−1 ≤ δ(f (c)) ≤ −k 2 , k = 0, 1, 2, . . . we let f ′ (c) = (f1 (c), f2 (c), . . . , fk (c), Hk+1 (c, − log2 δ(f (c)) − k), g(c), g(c), G(c, − log2 δ(f (c)) − k, ∗, ∗, ∗, . . . ). For c ∈ D, let f ′ (c) = f (c). It is left to the reader to show that f ′ is a Z-embedding which is ε-close to f . Proof of Theorem 4.1.1. Obviously, X ω ∈ M2 and X ω is the countable union of its strong Z-sets. Show that X ω is strongly M2 -universal. Since X ∈ AR, we see that X contains a topological copy of the unit segment and therefore X ω contains a topological copy of the Hilbert cube. Thus, M0 ⊂ M0 (X) and, by Lemma 4.1.2, X ω is strongly M0 -universal. The space X ω , being the countable union of its strong Z-sets, is strongly A1 -universal. In particular, X ω contains a closed topological copy of Σ. Since X ω ∼ = (X ω )ω , the space X ω ω ω contains a closed topological copy of Σ . Thus X is M2 -universal and, by Lemma 4.1.2, X ω is strongly M2 -universal. Surprisingly, but this positive result cannot be extended onto higher Borel classes. Let A be a dense subset of the Cantor cube C = {0, 1}ω . We regard C as a linearly independent subset of l2 (see Bessaga,Pelczynski [1975]). Put X = span C, Y = span A. ∑n For each y ∈ Y , y ̸= 0 there exists a unique representation y = i=1 λi ai with λi ∈ R \ {0} and ai ∈ A. In this case, the set supp(y) = {a1 , . . . , an } is called the support of y. We put supp(0) = ∅. Let An = {y∪∈ Y | | supp(y)| ≤ n}. It is easy to check that An are strong ∞ Z-sets in Y = n=1 An . The space Y can be of arbitrarily high Borel complexity. At the same time, the following theorem shows that Y ω is not Mξ -absorbing for ξ ≥ 3. 4.1.3. Theorem. The space Y ω is not A2 -universal. Proof. Assume the contrary. Then by Theorem 3.1.1 there exists an embedding h : (Qω , Qω \ Σω ) → (X ω , Y ω ), h = (hi ) where hi are the coordinate maps.

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127

We are going to construct a sequence (qi )∞ i=1 in Σ and a sequence of natural numbers n1 < n2 < . . . such that for each j ∈ N we have hj ({q1 } × · · · × {qnj } × Q × Q × . . . ) ⊂ Y . ∪∞ Let j = 1. Represent Y as Y = {0} ∪ k,l=1 Ykl where Ykl = {y = ∑l 1 1 i=1 λi ai | ai ∈ A, d(ai , aj ) ≥ k and |λi | ≥ k }. Obviously, Ykl are closed ω ω in Y . Since the set Q \Σ is of the second category and Qω \Σω ⊂ h−1 1 (Y ), there exists an open subset U ⊂ Qω \ Σω such that h−1 (Y ) ⊃ U for some kl 1 k, l ∈ N. Thus we can choose a nonempty connected open set in Qω of the form V = V1 × V2 × · · · × Vn1 × Q × Q × . . . , for some n1 , such that h−1 (Y ) ⊃ V \ Σω . It is easy to see that the restriction of the support map supp |Ykl : Ykl → exp C is continuous. Note that exp C is a 0-dimensional compactum (see Fedorchuk,Fillipov [1988]). Since the set V \ Σω = V ∩ (Qω \ Σω ) is connected, the image of the set V \ Σω under the map supp ◦h1 is a singleton consisting of the point {a1 , . . . , ai } ∈ exp C, ai ∈ A. Thus h1 (V \ Σω ) ⊂ span({a1 , . . . , ai }). Since span({a1 , . . . , ai }) is a closed subset in X, we have h1 (V ) ⊂ h1 (ClQ (V \ Σω )) ⊂ span({a1 , . . . , an }) ⊂ Y. Since Σ is dense in Q, we can choose points qj ∈ Vj ∩ Σ, j ≤ n1 . Then h1 ({q1 } × · · · × {qn1 } × Q × Q × . . . ) ⊂ Y. Since the pairs ({q1 } × · · · × {qn1 } × Q × Q × . . . , {q1 } × · · · × {qn1 } × Q × Q × · · · ∩ (Qω \ Σω )) and (Qω , Qω \ Σω ) are homeomorphic, we can proceed in this way and, ﬁnally, obtain a point q = (q1 , q2 , . . . ) ∈ Σω with h(q) ∈ Qω . This gives a contradiction. Exercises and Problems to §4.1.

∪

ω is not 1. Find for each ordinal ξ > 0 an absolute retract X ∈ / α

3 εk−1 4

for m < n.

Let B be a neighborhood of 1 in X such that each point of B can be joined with 1 by a path in B of diameter < 41 εk−1 . Since (B, d) fails to be totally bounded, there is an εk > 0 such that (4k ) holds and no compact set in X is an εk -net for B. Write gk (Q1 ) = f (Q1 ) and, if gk |Q1 ∪ · · · ∪ Qn−1 is already known, use the deﬁnition of εk to select a point p ∈ B with d(p, x) > εk for x ∈ {ab−1 | a, b ∈ gk (Q1 ∪ · · · ∪ Qn−1 ) ∪ gk−1 (Qn )}. Let h : I → X be a path such that h(0) = 1, h(1) = p, and diam h(I) < 14 εk−1 . We deﬁne gk |I n by gk (x) = h ◦ λ(x) · gk−1 (x) for x ∈ I n , where λ : D → I is a map such that λ(Dk \ U ) ⊂ {1} and λ(D \ Dk+1 ∪ Dk−1 ) ⊂ {0}. Since d is right-invariant, it follows from the choice of p that (3k ) holds for x ∈ Qn . Moreover, if x ∈ Qn ∩ Dk , y ∈ Qm and m < n then either x ∈ / U , in which case gk (x) = pgk−1 (x) and d(gk (x), gk (y)) = d(p, gk (y) · (gk−1 (x))−1 ) > εk , or else d(gk (x), gk (y)) > εk by (3k )–(5k ) and the triangle inequality. Thus (2k ) holds and by induction on n we deﬁne gk |Qn so that the resulting map gk : D → Y satisﬁes conditions (1k )—(4k ). By (1) and (3) there is a well-deﬁned map g = lim gk satisfying (6)

d(g(x), gk (x)) ≤

∑

d(gi+1 (x), gi (x))

0. Therefore, for such a sequence, there is k ∈ N with xi ∈ Dk for all i and, by (2) and (6), we get d(g(xi ), g(xj )) = d(gk (xi ), gk (xj )) > εk

for i < j,

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APPLICATIONS I

contrary to the assumed convergence of {g(xi )}. Proof of 4.2.1. We will prove a little bit more general result: every inﬁnitedimensional topological group that is a strongly LC ∞ -space satisﬁes SDAP (for the deﬁnition and properties of strongly LC ∞ -spaces see Ex.12 of §1.2). So, suppose G is an inﬁnite-dimensional topological group that is strongly LC ∞ , and let d be a right-invariant metric for G. In order to apply Lemma 4.2.2, we have to prove that no neighborhood of 1 in G is totally bounded in the metric d. Suppose, on the contrary, W is a neighborhood of 1 ∈ G, totally bounded in the metric d. Without loss of generality, W is symmetric (in place of W , we may consider the symmetric totally bounded neighborhood W ∩ W −1 ). Consider the metric d∗ (x, y) = d(x, y) + d(x−1 , y −1 ) on ˜ be the completion of the metric space (G, d∗ ). It is well known G, and let G (see A.Kechris [1995, p.58]) that the group operation of G can be extended ˜ so that G ˜ is a topological group containing G as a subgroup. over G We claim that the map id : (W, d) → (W, d∗ ) is a uniform homeomorphism. Since d∗ (x, y) = d(x, y) + d(x−1 , y −1 ), to show this it suﬃces to verify that the map x 7→ x−1 is uniformly continuous on (W, d). Given a symmetric neighborhood U of 1 in G, let {a1 , . . . , an } be points such that {U ai }i≤n covers W . Let V be a neighborhood of 1 such that ai V ⊂ U ai for i ≤ n. Given a ∈ W we then have, with i ≤ n taken so that x ∈ U ai , xV ⊂ U ai V ⊂ U 2 ai ⊂ U 3 x. Thus if y ∈ V −1 x, then y −1 ∈ U 3 x−1 . This implies id : (W, d) → (W, d∗ ) is a uniform homeomorphism and thus W is totally bounded in the metric ¯ of W in G ˜ is a compact neighborhood of d∗ . Consequently, the closure W ˜ 1 in G, and we get a contradiction with the following fact. ˜ is not locally compact. Claim. The group G Proof. By a result of Hoﬀman [1963], every locally compact strongly LC ∞ ˜ ⊃ G is inﬁnite-dimensional, to prove group is ﬁnite-dimensional. Since G ˜ is strongly LC ∞ . According our Claim, it suﬃces to show that the space G to Ex.14 to §1.2, for this it is enough to verify that the metric space (G, d∗ ) is uniformly strongly LC ∞ . Fix ε > 0. Since G is strongly LC ∞ , there is δ > 0 such that for every n ∈ N and every map f : ∂I n → B(1, δ) there exists a map f¯ : I n → B(1, ε/2) extending f . We claim that for every x ∈ G, every n ∈ N, and every map f : ∂I n → B(x, δ/2) there exists a map f¯ : I n → B(x, ε) extending f . To prove this, consider the map x−1 · f : ∂I n → G and notice that x−1 f (∂I n ) ⊂ B(1, δ). Indeed, taking any y ∈ f (∂I n ) ⊂ B(x, δ/2), we get d∗ (1, x−1 y) = d(1, x−1 y) + d(1, y −1 x) = d(y −1 , x−1 ) + d(x−1 , y −1 ) = 2d(x−1 , y −1 ) ≤ 2(d(x, y) + d(x−1 , y −1 )) = 2d∗ (x, y) < 2 2δ = δ. By the choice of δ, the map g = x−1 f : ∂I n → B(1, δ) can be extended to a map g¯ : I n → B(1, ε/2). Then the map f¯ = x · g¯ : I n → G extends f

4.2. TOPOLOGICAL GROUPS

131

and has the property f¯(I n ) ⊂ B(x, ε). Thus (G, d∗ ) is uniformly strongly LC ∞ . B. The strongly universal property in ANR-groups. Below by Pt we denote the class of all one-point spaces. 4.2.3. Theorem. Let C be a 2ω -stable M1 -hereditary Pt-additive weakly A1 -additive compactiﬁcation-admitting class of spaces. An inﬁnite-dimensional ANR-group G is strongly C-universal if and only if G is C-universal. Proof. Suppose G is an inﬁnite-dimensional C-universal ANR-group. As we have already noticed in the proof of 4.2.1, the group G can be considered ˜ Let d be a as a subgroup of a complete-metrizable topological group G. ˜ Since G is an ANR, by Theorem 1.2.4, there is right-invariant metric on G. ˜ such that M is an ANR containing G as a homotopy a Gδ -subset M ⊂ G dense subset. Now we are going to show that the pair (M, X) is strongly (M0 ∩ C, C)-preuniversal. For that we need ˜ of 1, every pair (K, C) ∈ 4.2.4. Lemma. For every neighborhood U ⊂ G (M0 ∩ C, C), and every point c0 ∈ C there is an embedding f : K → U such that f (c0 ) = 1 and f −1 (G) = C. ˜ = {c0 } ∪ (K \ {c0 } × N) be the one-point compactiﬁcation of Proof. Let K ˜ It follows the space K \ {c0 } × N, and let C˜ = {c0 } ∪ C \ {c0 } × N ⊂ K. ˜ ˜ from the properties of C that (K, C) ∈ (M0 ∩ C, C). Since the space G ˜ G) is (M0 ∩ C, C)-universal; see Theorem 3.1.1. is C-universal, the pair (G, ˜ →G ˜ with e−1 (G) = C. ˜ Without loss Hence there is an embedding e : K of generality, e(c0 ) = 1 (for otherwise, in place of e, we may consider the embedding e(c0 )−1 e). Find n0 ∈ N such that e((K \ {c0 }) × {n0 }) ⊂ U , and deﬁne a required embedding f : K → U by { 1, if x = c0 ; f (x) = e(x, n0 ), if x ̸= c0 . Now let us return to the proof of strong (M0 ∩ C, C)-preuniversality of the pair (M, X). Fix a cover U ∈ cov(M ), a pair (K, C) ∈ (M0 ∩ C, C), a closed subset B ⊂ K, and a map f : K → M such that (f |B)−1 (X) = B ∩ C. Since G is homotopy dense in M , we can additionally assume that f (K \ B) ⊂ G. By compactness of f (K), there exists an ε > 0 such that {B(x, ε)}x∈f (K) ≺ U. ˜ = Let A be a countable dense set in 2ω . Consider the quotient space K ˜ We identify (K × 2ω )/(B × 2ω ) and the quotient map π : K × 2ω → K. ˜ with the one-point compactiﬁcation {∗} ∪ (K \ B × 2ω ) of the the space K space K \ B × 2ω . Consider the subset C˜ = {∗} ∪ K \ B × A ∪ C \ B × 2ω

132

APPLICATIONS I

˜ It follows from the properties of C that (K, ˜ C) ˜ ∈ (M0 ∩ C, C). By in K. ˜ Lemma 4.2.4, there is an embedding e : K → B(1, ε) such that e(∗) = 1 ˜ and e−1 (G) = C. ˜ by g(z) = (e◦π(z))·(f ◦prK (z)), z ∈ K ×2ω Deﬁne a map g : K ×2ω → G (here prK : K × 2ω → K denotes the projection). Notice that g is ε-close to f ◦ prK , g|B × 2ω = f ◦ prK |B × 2ω , and (g|K \ B × 2ω )−1 (G) = K \ B × A ∪ C \ C × 2ω = C˜ \ {∗}. Then (K \ B × 2ω ) \ g −1 (M ) is a σ-compact set in K × 2ω contained in K × 2ω \ A. Since 2ω \ A is not σ-compact, there is t0 ∈ 2ω \ A such that (K \ B) × {t0 } ⊂ g −1 (M ). Deﬁne ﬁnally a map f¯ : K → M by f¯(k) = g(k, t0 ), k ∈ K. It is easily veriﬁed that f¯|B = f |B, d(f¯, f ) < ε and f¯−1 (G) = C. Therefore, the pair (M, X) is strongly (M0 ∩ C, C)preuniversal, and by Theorem 3.2.16, it is strongly (M0 ∩ C, C)-universal. By Theorem 4.2.1, G satisﬁes SDAP, so we can apply Theorem 1.7.9 to conclude that the space G is strongly C-universal. 4.2.5. Theorem. For every 0 ≤ n ≤ ∞, an inﬁnite-dimensional ANRgroup G is strongly M1 [n]-universal if and only if G is M1 [n]-universal. Proof. Suppose G is an inﬁnite-dimensional M1 [n]-universal ANR-group. By Theorem 4.2.1, the space G satisﬁes SDAP. To show that G is strongly M1 [n]-universal, we will apply Theorem 3.2.19. Fix a right-invariant metric d on G, an open set U ⊂ G, a cover U ∈ cov(U ), and a map f : C → U , where C ∈ M1 [n]. Let V ∈ cov(U ) be a cover with StV ≺ U. Since U is an ANR, there are a locally compact space K and two maps p : U → K, q : K → U such that (q ◦ p, id) ≺ V. The set U , being open in G, satisﬁes SDAP. Hence, by 1.3.4, we can additionally assume that the map q is perfect, and consequently, q(K) is a closed locally compact set in U . Consider the map f ′ = q ◦ p ◦ f : C → U and notice that (1)

(f, f ′ ) ≺ V.

Let W ∈ cov(U ) be a locally ﬁnite cover such that W ≺ V and for every x ∈ U the intersection ClX (St(x, W)) ∩ q(K) is compact. Pick up a continuous function ε : q(K) → (0, 1] such that {B(x, 2ε(x)}x∈q(K) ≺ W. Let M = s, if n = ∞, and M be the N¨obeling space of dimension dim C, if n ̸= ∞. Since the space G is M1 [n]-universal, there is a closed embedding M ⊂ G. Since G is homogeneous, we may assume that 1 ∈ M . ∪k 1 For every k ∈ N let Ck = (ε ◦ f ′ )−1 ([ k+1 , k1 ]), and C˜k = i=1 Ci .

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133

Claim. There is a closed embedding e : C → M \ {1} such that for every k ∈ N we have 0 < inf{d(e(c), 1) | c ∈ Ck } ≤ sup{d(e(c), 1) | c ∈ Ck }

n and 0 ≤ xi ≤ 1 if 1 < i ≤ n}

n=1

of l2 . The compactifying point is denoted by y0 . It is elementary to check that ind Y = ω0 . By 2.2.18, there is an everywhere Y -universal compactum K with ind K = ω0 . Now Theorem 4.2.11 follows from the following statement. 4.2.12. Proposition. No space X ∈ σM[ind, ω0 ] admits a cancellative operation f : K × I → X. Proof. Suppose on the contrary that f : K × I → X is a cancellative ∪∞ operation and X ∈ σM[ind, ω0 ]. Write X = n=1 Xn , where each Xn ∈ M[ind, ω0 ] is closed in X. By Baire Theorem, one of f −1 (Xn ) has nonempty interior in K × I. Since K is everywhere Y -universal and I is locally self-similar, we may assume f (Y × I) ⊂ Xn , and thus ind f (Y × I) ≤ ω0 . Since the operation f is cancellative, f (y0 , 0) ̸= f (y0 , 1). To get a contradiction, we will show that f (y0 , 0) and f (y0 , 1) cannot be separated in f (Y × I) by a ﬁnite-dimensional closed subset. Suppose converse, and let C ⊂ f (Y × I) be a closed subset of dimension n < ∞, separating the points f (y0 , 0) and f (y0 , 1). Then the set C ′ = f −1 (C) separates the points (y0 , 0) and (y0 , 1) in Y × I. Consider the map g = prI |C ′ : C ′ → I. Since for each t ∈ I the map f homeomorphically embeds the set Y × {t} into f (Y × I), we have dim(C ′ ∩ Y × {t})) ≤ n and hence, dim g ≤ n (recall that the dimension of a map is the supremum of dimensions of its ﬁbers). By Hurewicz formula (see, e.g. P.Aleksandrov, B.Pasynkov [1975]), dim C ′ ≤ dim I + dim g = n + 1. However, there exists a topological copy D of I n+3 in Y × I which contains the points (y0 , 0) and (y0 , 1) so that they can not be separated in D by any set of dimension ≤ n + 1.

136

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Exercises and Problems to §4.2. 1. Show that the absorbing space Ω(C) from Theorem 2.4.6 admits no cancellative operation Ω(C) × Ω(C) → Ω(C), but admits a cancellative operation Ω(C) × σ → Ω(C). 2. Show that an ANR-space X carrying a topological group structure is an M1 (ω)absorbing space if and only if X ∈ M21 (s.c.d.) \ M1 is an M1 (ω)-universal space. ˜ be a complete-metrizable ANR-group and G a subgroup in G ˜ such that G is 3. Let G ˜ Show that if G is an M1 -universal space then a homotopy dense Fσ -subset in G. ˜ G) is an (s × s, s × σ)-manifold. (In fact, there is an open embedding the pair (G, ˜ → s × s with e−1 (s × σ) = G). e:G 4. Show that no topological group is a Q-manifold. We say that a pointed space (X, x0 ) is C-universal, where C is a class of spaces, if for every C ∈ C and every c0 ∈ C there exists a closed embedding e : C → X with e(c0 ) = x0 . 5. Suppose that either C = M1 [n] for 0 ≤ n ≤ ∞, or C is a 2ω -stable M1 -hereditary Pt-additive weakly A1 -additive compactiﬁcation-admitting class of spaces. Let G be a topological group and let X be a multiplicative subset in G such that X is an ANR with SDAP. a) Show that X is strongly C-universal provided it contains a closed in G subset D ∋ 1 such that the pointed space (D, 1) is C-universal. b) Assuming X to be closed in G, show that X is strongly C-universal if and only if the pointed space (X, 1) is C-universal. c) Suppose that X is closed in G and X is a homogeneous space. Show that X is strongly C-universal if and only if X is C-universal. 6. Show that every analytic space carrying a topological group structure is either completemetrizable or of the ﬁrst Baire category. Hint: Use the fact that each analytic space X can be written as X = G ∪ A, where G ∈ M1 and A is a meager set in X. 7. Find a Baire space X ∈ / M1 that carries a topological group structure. Hint: see 5.5.2. 8. Find an example of an AR-group G ∈ A3 \ M1 that is not a Zσ -space. Hint: consider the group W (R, Q). 9. (Open Problem) Let G ∈ A1 (s.c.d.) be an inﬁnite-dimensional locally contractible topological group. Is G a σ-manifold? Is G a σ-manifold if, additionally, G is M0 (ω)universal? 10. (Open Problem) Suppose G is a σ-compact ANR-group that contains a topological copy of the Hilbert cube. Is G a Σ-manifold? 11. Suppose G is an inﬁnite-dimensional Polish ANR-group. Show that G contains a subgroup H such that the pair (G, H) is locally homeomorphic to (s, Σ). 12. (Open Problem) Suppose G is an inﬁnite-dimensional Polish ANR-group. Does G contain a subgroup H such that the pair (G, H) is locally homeomorphic to (s, σ)? to (s × s, σ × s)? 13. (Open Problem) Homeomorphism groups. For a compactum K let H(K) denote the homeomorphism group of K, endowed with the compact-open topology. For a subset C ⊂ K let H(K, C) = {h ∈ H(K) | h(C) = C}. Suppose (Q, X) is an absorbing pair. Is the pair (H(Q), H(Q, X)) absorbing? If so, for which class? To be more concrete, is the pair (H(Q), H(Q, Σ)) homeomorphic to (Q × s, Π2 × s)? Remark that by S.Ferry [1977], the homeomorphism group H(Q) is a completemetrizable AR, and hence H(Q) is homeomorphic to s. 14. (Hartman-Mycielski Construction) Let X be a space and HM (X) = {f : [0, 1) → X |

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there is a sequence 0 = t0 < t1 < · · · < tk = 1 such that f |[ti−1 , ti ) is a constant map for each i = 1, . . . , k} (note that f needs not be continuous in this deﬁnition). If d is a bounded metric on X, then HM (X) is endowed with the metric d∗ : d∗ (f, g) = ∫1 d(f (t), g(t))dt. 0

a) Show that the topology on HM (X) generated by d∗ does not depend on the choice of admissible bounded metric d. b) Show that the map X → HM (X) deﬁned by x 7→ fx ≡ x is a closed isometric embedding. Further we will identify X with its image in HM (X). c) Show that for every space A, every map f : B → X deﬁned on a closed subspace B of A can be extended to a map f¯ : A → HM (X). Hint: Use the “convex” structure on HM (X) generated by the map ξ : HM (X) × HM (X) × I → HM (X): ξ(f, g, t)(τ ) = f (τ ) if τ < t and g(τ ) otherwise. d) Show that HM (X) is an AR. Hint: Note that HM (X) is a retract of HM (HM (X)) and then apply c). e) Suppose X is topological group (linear topological space). Then the operations on X have their pointwise extensions onto HM (X). Show that HM (X) is a topological AR-group (linear topological AR-space) with respect to these operations and X is a closed subgroup (subspace) in HM (X). e) Suppose X ∈ A1 (s.c.d), |X| > 1. Show that HM (X) is homeomorphic to σ. f) Show that for every topological σ-compact ANR-group G there is a topological group H and a group homomorphism h : H → G such that h is an A1 (s.c.d.)-soft map, and H is a σ-manifold.

§4.3. Hyperspaces In this section we investigate topology of hyperspace exp X. Recall that the hyperspace exp X of a space X is the space of all nonempty compact subsets, endowed with the Vietoris topology. The base of this topology consists of the sets ⟨V1 , . . . , Vn ⟩ = {A ∈ exp | A ⊂

n ∪

Vi and A ∩ Vi ̸= ∅ for each i}

i=1

where V1 , . . . , Vn are open subsets of X. If d is an admissible metric on X then the Vietoris topology on exp X is generated by the Hausdorﬀ metric dh (A, C) = inf{ε > 0 | A ⊂ Bε (C) and C ⊂ Bε (A)}. A continuum is, by deﬁnition, a connected compactum. 4.3.1. Definition. A space X is continuum-connected if each pair of point in X is contained in a subcontinuum of X. A space X is called locally continuum-connected if there exists a base of continuum-connected sets. It is known that for complete spaces the notions of locally connectedness locally continuum-connectedness and locally arcwise connectedness coincide. The continuum X is called Peano continuum if X is locally connected.

138

APPLICATIONS I

It is known that exp X ∈ ANR(AR) iﬀ X is locally continuum-connected (connected and locally continuum-connected) (see Curtis [1980]). When X is a locally continuum-connected space we can choose on X a metric d satisfying the property (∗) for each points x, y ∈ X if d(x, y) < ε then there exists a subcontinuum K of X such that diam K < ε and {x, y} ⊂ K. To obtain that metric put d(x, y) = inf{diamd0 (K) | K ⊂ X is a subcontinuum of X which contains x and y} where d0 is any metric on X. Further we consider locally continuum-connected spaces with metrics satisfying condition (∗). Let C be one of the following classes of spaces: compact, locally compact, complete, coanalytic, nowhere locally compact, and nowhere complete. One can check the next proposition: 4.3.2. Proposition. X ∈ C iﬀ exp X ∈ C.

We start with investigation of the topology of exp X for locally compact locally connected X. Let d be a metric on X. We can assume that a closed subset A ⊂ X is compact if and only if diam A < ∞. Deﬁne the homotopy D : exp X ×[0, 1] → exp X by the formula D(A, t) = {x ∈ X | d(x, A) ≤ t}. One can check that D is continuous. 4.3.3. Theorem. The space exp X is a Q-manifold iﬀ X is locally compact and locally connected. Proof. We will use Characterization Theorem 1.1.23. Let K be a compactum, f : K → exp X a map and U ∈ cov(exp X). We can assume that U consists of ε-balls. Deﬁne a map f ′ : K → exp X as follows f ′ (a) = D(f (a), ε/2) for a ∈ K. Evidently, f ′ is U-close to f . Put Z = ∪f ′ (K). It is easy to see that Z is a compactum. Let {Bi }ni=1 be a ﬁnite ∪n cover of Z with open sets Bi such that diam Bi ≤ ε/6. Then f ′ (K) = i=1 Ai where Ai = {A ∈ exp X | A ⊃ Bi }. To complete the proof we need to show that for every open set O ⊂ X with compact closure the set A = {A ∈ exp X | A ⊃ O} is a Z-set in X. Let g : Q → X be a map, and U a cover of X. We can assume that U consists of δ-balls. Let s ∈ O be a point. Put γ = min{δ/6; d(s, X \ O)/2} and O1 = X \ B(s, γ). Deﬁne the function g ′ : K → exp X by the formula g ′ (A) = D(A, γ) ∩ O1 . Evidently g ′ is U-close to g and g ′ (K) ∩ A = ∅. Conversely, if exp X is a Q-manifold, thus an ANR, then X must be locally connected. Since X admits a closed embedding into exp X, X must be locally compact. Theorem 4.3.3 implis: 4.3.4. Corollary. The space exp X is homeomorphic to Q iﬀ X is a Peano continuum.

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139

4.3.5. Theorem. The space exp X is homeomorphic to Q \ {∗} iﬀ X is a connected locally connected locally compact noncompact space. We will need the following lemma: 4.3.6. Lemma. Let X be as in Theorem 4.3.5 and O ⊂ X an open subset. Then the set O− = {A ∈ exp X | A ∩ O ̸= ∅} is contractible. Proof. Fix a contracting homotopy H : exp X × I → exp X with H0 = id and H1 ≡ const and ﬁx any point o ∈ O. Deﬁne the function m : exp X → R by the formula m(A) = min{d(o, a) | a ∈ A}. One can easily check that the function m is continuous. Deﬁne the homotopy G : O− × [0, 1] → O− by the formula 1 D(A, 2tm(A))), if 0 ≤ t ≤ , 2 G(A, t) = H(D(A, m(A)), 2t − 1) ∪ {o}, if 1 ≤ t ≤ 1. 2 Proof of Theorem 4.3.5. Let K ∪1∞⊂ K2 ⊂ . . . be a sequence of compacta such that Ki ⊂ Int Ki+1 and i=1 Ki = X. By Lemma 4.3.6, (X \ Ki )− ˜ = {∗} ∪ exp X be the one-point compactiﬁcation of is contractible. Let X exp X. Then the family {(X \ Ki )− }∞ i=1 is the neighborhood base at ∗. It follows from Theorem 4.3.2 that exp X is a Q-manifold. By Ex.15 to §1.2, ˜ is homeomorphic to Q and exp X to Q \ {∗}. X Conversely, if exp X is homeomorphic to Q \ {∗}, by Theorem 4.3.2, the space X must be locally connected and locally compact. Since Q\{∗} ∈ AR, X is connected. Evidently, X is noncompact. Now we consider the spaces X whose hyperspaces are homeomorphic to l2 . Theorem 4.3.7. The space exp X is homeomorphic to l2 if and only if X is connected, locally connected, complete and nowhere locally compact. Proof. Suppose exp X ∼ = l2 . Since l2 is a complete nowhere locally compact AR, X satisﬁes the conditions of the theorem. Conversely, suppose X satisﬁes the hypotheses. Then exp X is a complete AR, and it remains to verify that exp X satisﬁes SDAP. n Let a map f : ⊔∞ i=1 Ki → exp X, where each Kn = I is the n-dimensional cube, and an open cover U of exp X be given. A map g : ⊔∞ i=1 Ki → exp X must be constructed such that f and g are U-close and {g(Kn )} is a discrete family in exp X. Let d be a metric on X and dH the respective Hausdorﬀ metric on exp X. For A ∈ exp X and µ > 0, let N (A; µ) = {x ∈ X | d(x, A) < µ}, and ˜ (A; µ) = {F ∈ exp X | dH (A, F ) < µ}. N

140

APPLICATIONS I

There exists a map µ : exp X → (0, ∞) such that, for each A ∈ exp X, ˜ (A; µ(A)) is contained in some element of U. Thus f, g : ⊔∞ Ki → exp X N i=1 will be U-close provided dH (f (y), g(y)) < µf (y) for each y ∈ ⊔∞ i=1 Ki . We can also choose a map η : exp X → (0, ∞) such that if x ∈ N (A; η(A)) there exists an arc J in X connecting x and A, with diam J < 21 µ(A). Necessarily, η(A) ≤ 21 µ(A). Subdivide each cube Kn so that the given map f : ⊔∞ i=1 Ki → exp X satisﬁes the following conditions for each simplex σ: (i) diam f (σ) < 12 inf σ ηf , (ii) supσ ηf < 2 inf σ ηf , (iii) supσ µf < 32 inf σ µf . For each n, let Vn be a locally ﬁnite (therefore countable) open cover of X with mesh Vn < n1 . In each element of Vn choose an inﬁnite discrete subset of X. Then the union of such subsets over all the elements of Vn is an inﬁnite discrete subset Z(n) of X. We deﬁne g ﬁrst at each vertex of the complex K1 . For a vertex v let nv be the smallest integer such that 1/nv < 41 ηf (v), and choose a point zv in Z(nv ) such that d(zv , f (v)) < 1/nv . Set g(v) = f (v) ∪ {zv }. (1) We now extend g over the 1-skeleton K1 of K1 . Let τ be an edge of K1 , with the endpoints v and w and the midpoint m. Since d(zv , f (m)) ≤ d(zv , f (v)) + dH (f (v), f (m)) ≤ 41 ηf (v) + 12 inf τ ηf ≤ 21 inf τ ηf + 21 inf τ ηf ≤ ηf (m), there exists an arc Jv in X connecting zv and f (m), with diam Jv < 1 2 µf (m). Likewise, there exists an arc Jw connecting zw and f (m), with diam Jw < 12 µf (m). Deﬁne g(m) = f (m)∪Jv ∪Jw . Then dH (f (m), g(m)) ≤ 1 2 µf (m). Let h : I → exp X be a path from f (m) to g(m) such that h(0) = f (m), h(1) = g(m), and f (m) ⊂ h(t) ⊂ g(m) for each t. Deﬁne g on the segment [vm] (and similarly on the segment [wm]) as follows: 1 f ((1 − 2t)v + 2tm) ∪ {zv }, if 0 ≤ t ≤ , 2 g((1 − t)v + tm) = 1 h(2t − 1) ∪ {zv }, if ≤ t ≤ 1. 2 Thus g is deﬁned over each edge τ = [vm] ∪ [wm]. One can check that for each p ∈ τ we have dH (f (m), g(p)) ≤ 21 µf (m). (1)

The map g is deﬁned similarly over each 1-skeleton Ki of complex Ki with the stipulation that the points zv are chosen from Z(nv ) \ ∪{g(y) | y ∈ K1 ∪ · · · ∪ Ki−1 }. Extend g over all of Ki as follows. Let Ci denote (1) the hyperspace of subcontinua of the 1-skeleton Ki . There exists a map (1) r : Ki → Ci such that f (p) = {p} for each p ∈ Ki , and such that if σ is the carrier of the point y, then r(y) ⊂ σ (1) . (First deﬁne r over the

4.3. HYPERSPACES

141

(2)

2-skeleton Ki as follows. Consider each 2-simplex σ as the cone over its boundary σ (1) , with cone point c. For each point y = tc + (1 − t)p in σ where p ∈ σ (1) take r(v) to be a subcontinuum of σ (1) centered at p and with length proportional to t. If t = 1, then r(y) = r(c) = σ (1) . Similarly, r is extended inductively over the higher-dimensional skeletons.) Now deﬁne g(y) = ∪{g(p) | p ∈ r(y)}. One can check that dH (f (y), g(y)) ≤ µf (y) for each y ∈ Ki . Thus g : ⊔∞ i=1 Ki → exp X is U-close to f . Replacing g(y) by f (y) ∪ g(y), we may assume also that f (y) ⊂ g(y) for each y. It is left to the reader to check that {g(Ki )} is a discrete family in exp X. Finally, we investigate the space exp X for coanalytic nowhere complete X. In the remaining part of this section X will denote a coanalytic nowhere complete connected locally connected space with a metric d satisfying condition (∗). Denote by Y the completion of X. The metric on Y is denoted by the same letter d. By ϱ we denote the Hausdorﬀ metric on exp X generated by d. Without loss of generality we can assume that Y is nowhere locally compact. The proof of the following lemma has technical character and is left to the reader. 4.3.8. Lemma. For each ε > 0 and x, y ∈ Y with d(x, y) < ε there exists a continuum K ⊂ X ∪ {x, y} containing {x, y} with diam K < ε. Thus Y is a connected locally connected nowhere locally compact complete space and, by Theorem 4.3.7, exp Y is homeomorphic to the Hilbert space l2 . 4.3.9. Lemma. The space exp X is homotopy dense in exp Y . Proof. Let d be a metric on X with property (∗). Let f : I k → exp Y be any map with f (∂I k ) ⊂ exp X and ε > 0. We have to ﬁnd an ε-close to f map f˜ : I k → exp X with f˜|∂I k = f |∂I k . Fix any metric d1 on I k and choose a triangulation N of the interior k I \ ∂I k such that for each simplex σ ∈ N 1 εσ , 6 { } where εσ = min ε, inf{d1 (x, ∂I k ) | x ∈ St(σ)} . Since X is dense in Y , for each vertex v ∈ N (0) we can choose a ﬁnite set Fv ⊂ X such that (1)

diam f (σ)

1, c) l2 iﬀ X is not locally compact; (Hint: Use the fact that cc X is a perfect retract of exp X.) 7. Let cp(Rn ), n > 1 denote the subspace of cc(Rn ) consisting of all convex polyhedra (i.e. convex hulls of ﬁnite sets). Prove that (cc(Rn ), cp(Rn )) ∼ = (Q \ {∗}, σ \ {∗}). 8. (Open problem) Find naturally deﬁned subspaces X of cc(Rn ) for which (cc(Rn ), X) ∼ = (Q \ {∗}, Λα (s.c.d.) \ {∗}) (resp. (Q \ {∗}, Λα (s.c.d.) \ {∗})). 9. For each A ∈ cc(Rn ) with Int(A) ̸= ∅ let S(A) = {x ∈ A | there exist distinct y1 , y2 ∈ Bd A such that d(x, Bd A) = d(x, y1 ) = d(x, y2 )} (d is a usual Euclidean metric). Let S = {A ∈ cc(Rn ) | Int A ̸= ∅ and Cl S(A) = A}. Prove that (cc(Rn ), S) ∼ = (Q \ {∗}, s \ {∗}). Hint. See Bazylevych [1991]. 10. A set A ∈ cc(Rn ) with Int A ̸= ∅ is called strictly convex if Bd A contains no straight segments. Let C denote the subset of cc(Rn ) consisting of all strictly convex compacta. Prove that (cc(Rn ), C) ∼ = (Q \ {∗}, s \ {∗}). Hint: See Bazylevych [1993].

146

APPLICATIONS I

11. A convex set X is called measure-convex, provided for every compact subset K ⊂ X the closed convex hull conv(K) is compact. Suppose X is a measure-convex set of a Fr` echet space. a) Show that cc X is a perfect retract of exp X, b) Suppose X is nowhere topologically complete. Show that cc X is strongly P2 universal. ∼ Π2 . c) Suppose X ∈ P2 is of the ﬁrst Baire category. Show that cc X = 12. (Open Problem) Let X = {A ∈ exp Q × Q | pr1 |A : A → Q is a (n)-soft map }. Describe the topology of X . ˜ (I) = {f : I → I | f is a monotone left-semicontinuous function}. We will Let M ˜ (I) with E(f ) = {(x, y) ∈ I × I | y ≥ f (x)} ∈ exp I × I and endow identify each f ∈ M ˜ (I) with the topology inherited from exp I × I. M ˜ ∼ ˜ (I), M ˜ (I) ∩ C(I, I)) ∼ 13. Prove that M = Q and (M = (Q, s). Hint: See Tkach [1996]. ˜ (I) | f ′ ≡ 0 on an open set of Lebesgue measure 1}. Prove that 14. Let H = {f ∈ M ˜ (I), H) ∼ (M = (Qω , Σω ). Hint: See Tkach [1996].

Historical Notes and Comments to Chapter IV §4.1. Theorem 4.1.1 is due to T.Dobrowolski and J.Mogilski [1992]. They have posed the question about a counterpart of this result related higher Borelian classes. The negative answer to this question (Theorem 4.1.3) is due to Banakh and Cauty [2001]. The results concerning Fr`echet products are due to T.Radul. §4.2 Theorem 4.2.1 is taken from T.Dobrowolski [1986b], Lemma 4.2.2 and Theorem 4.2.6 is due to T.Dobrowolski and H.Toru´ nczyk [1981]. The results of §4.2.D belong to M.Zarichnyi. All other results of §4.2 belong to T.Banakh. §4.3. Corollary 4.3.4 is due to Curtis, Schori [1978], Theorem 4.3.5 and Theorem 4.3.7 are due to Curtis [1979,1980]. The proof of Theorem 4.3.7 is taken from Curtis [1979]. All results concerning noncomplete case are due to Banakh and Cauty [1997b].

5.1. LOCALLY COMPACT CONVEX SETS

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Chapter V Applications II: Convex Sets

In this chapter we deal with the problem of topological classiﬁcation of convex sets in linear metric spaces, which traces its history back to Banach and Fr´echet. A brilliant exposition of the situation in this area on the state up to 1975 can be found in the book of C.Bessaga, A.Pelczy’nski [1975]. Since then a signiﬁcant progress in the subject was made and our aim is to describe the present situation in this realm. Below under a (closed) convex set we mean a (closed) convex subspace of a linear metric space. All consistent results on topology of convex sets are proved for convex sets that are absolute retracts. In fact, the problem of ﬁnding a convex sets, which is not an AR, stood more than 60 years, and was solved quite recently by R.Cauty [1994b], who constructed a σ-compact linear metric space that is not an AR. We start (in §5.1) with the topological classiﬁcation of locally compact (closed) convex absolute retracts. Further, in the section §5.2, we inspect the strong discrete approximation property in convex sets and on this base, obtain a complete topological classiﬁcation of complete-metrizable (closed) convex sets whose completion either is non-locally compact or is an AR. Subsections §§5.3–5.4 are devoted to incomplete convex sets. Our strategy here is to show that a given convex set is a (co)-absorbing space in order to apply the Uniqueness Theorem 1.6.5. In §5.5 we present three important counterexamples: 1) Mill’s example of a pre-Hilbert space X, which is not strongly universal, 2) Marciszewski’s example of a normed absorbing space, which is homeomorphic to no convex set in l2 , and 3) Banakh’s example of a Borel pre-Hilbert space, which is not a Zσ -space. The obtained results are applied in §5.6 to probability measure spaces and in §5.7 to the function spaces Cp (X) and Cp∗ (X). §5.1. Locally compact convex sets In what follows, all metrics on linear metric spaces are assumed to be invariant and monotone. Recall that a metric d on a linear metric space L is called invariant if d(x + z, y + z) = d(y, z) for all x, y, z ∈ X, and monotone, provided d(t · x, 0) ≤ d(x, 0) for t ∈ [−1, 1], x ∈ X. For such a

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metric d let ∥x∥ = d(x, 0). For a convex set X in a linear metric space L denote by cc(X) = {y ∈ L | ∃x ∈ X with x + [0, ∞) · y ∈ C} the characteristic cone of X. Let us recall the topological classiﬁcation of ﬁnite-dimensional closed convex sets. Suppose X is a ﬁnite-dimensional closed convex set. • If cc(X) is a linear space ({0} is a linear space!), then X is homeomorphic to [0, 1]n × (0, 1)m , where m = dim cc(X), and n = dim X − m; • if cc(X) is not a linear space then X is homeomorphic to [0, 1]n × [0, 1), where n = dim X − 1. In this subsection we extend this classiﬁcation over all locally compact convex absolute retracts. 5.1.1. Theorem. Every locally compact inﬁnite-dimensional convex set X ∈ AR is a Q-manifold. Proof. We will apply Characterization Theorem 1.1.23. Fix n ∈ N, two maps f, g : I n → X, and ε > 0. First, we will prove that the maps f, g can be approximated by maps whose ranges lie in a ﬁnite-dimensional linear ε subspace. Let N be a triangulation of I n with diam f (σ) < 2(n+1) for σ ∈ ∑ ∑ ∞ n ′ N . Put f (q) = i=0 ti f (vi ) for q = i=0 ti vi a point of a∑ closed n-simplex n in N with vertices v1 , . . . , vn . Then ∥f (q) − f ′ (q)∥ ≤ i=0 ∥ti (f (vi ) − ε ε ′ n f (q))∥ < (n + 1) 2(n+1) = 2 . Analogously, ﬁnd a map g : I → X such that d(g ′ , g) < ε and span g ′ (I n ) is ﬁnite-dimensional. Then L0 = span(f ′ (I n ) ∪ g ′ (I n )) is a ﬁnite-dimensional linear subspace, and hence, there is a point x0 ∈ X\L0 . Letting f ′′ (q) = 2ε x0 + (1 − 2ε )f ′ (q), we obtain an ε-close to f map such that f ′′ (I n ) ∩ g ′ (I n ) = ∅. 5.1.2. Corollary. Every compact inﬁnite-dimensional convex set X ∈ AR is homeomorphic to the Hilbert cube Q. The following theorem supplies us with the topological classiﬁcation of inﬁnite-dimensional closed convex AR-sets. 5.1.3. Theorem. Let K ∈ AR be an inﬁnite-dimensional locally compact closed convex set. a) If cc(K) = {0} then K is homeomorphic to Q; b) if cc(K) is a linear space then K ∼ = Q × Rn , where n = dim cc(K); c) if cc(K) is not a linear space then K ∼ = Q × [0, 1). Proof. Suppose K is a convex subset of a linear metric space E. a) If cc(K) = {0}, then K is bounded and consequently, it is compact, see C.Bessaga, A.Pelczy´ nski [1975, p.99]. Then by 5.1.2, K is homeomorphic to Q. b) If cc(K) = E0 is a linear subspace of E, then E0 , being a closed subset of K, is locally compact. Then, E0 is ﬁnite-dimensional. By Michael Theorem, see C.Bessaga, A.Pelczy´ nski [1975], the coset map χ : E → E/E0

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admits a cross-section φ : E/E0 → E. Consider a homeomorphism Φ : E → E/E0 ×E0 given by Φ(x) = (χ(x), x−φ◦χ(x)). Φ maps the set K = K +E0 onto K ′ × E0 , where K ′ = χ(K) is an inﬁnite-dimensional convex set with cc(K ′ ) = {0}. To see that K ′ is locally compact and closed in E/E0 , notice that φ(E/E0 ) is a closed subset of E. Hence φ(K ′ ) = K ∩ φ(E/E0 ) is a locally compact closed subset of φ(E/E0 ). Since φ is a homeomorphism, K ′ is a locally compact closed subset in E/E0 , and thus by a), K ′ ∼ = Q. Then K is homeomorphic to K ′ × E0 ∼ = Q × Rn , where n = dim E0 = dim cc(K). c) Suppose cc(K) is not a linear space. Denote by K∞ the one-point compactiﬁcation of K and by ∞ the compactifying point. Since Q\{pt} is homeomorphic to Q × [0, 1), it is enough to show that K∞ ∼ = Q. For this we will apply Ex.15 to §1.2. Since cc(K) is not a linear space, there is a line ℓ ⊂ E such that K ∩ ℓ = x0 + [0, ∪∞) · x for some x0 ∈ E. To every set A ⊂ L let us assign the set A− = a∈A (a − [0, ∞) · x) ∩ K. It is easy to verify that whenever A is compact, A− is also compact. Since ∪∞ K is locally compact, it can be written as K = n=1 Kn , where each Kn ⊂ Int Kn+1 is compact. Then every Kn− is also compact. Moreover, passing to a suitable subsequence, if necessary, we may assume that Kn− ⊂ Int Kn+1 . Clearly, {K∞ \Kn− }n∈N forms a neighborhood base at ∞. Let us show that each Un = K\Kn− is an AR. Since Un is an ANR, it suﬃces to show that every map f : S k → Un from a k-dimensional sphere is homotopically trivial. Arguing as in Theorem 5.1.1, we see that f is homotopic to a map f ′ : S k → Un such that E0 = span({x0 } ∪ f ′ (S k )) is ﬁnite-dimensional. Let x∗ : E0 → R be any linear functional such that x∗ (x0 ) = 1. Since the set f ′ (S k )∪Kn− is compact, there are numbers m, M such that m < x∗ (y) < M for every y ∈ f ′ (S k )∪Kn− . Then the map f ′′ : S k → Un , deﬁned by f ′′ (y) = f ′ (y) + (M − m) · x0 for y ∈ S k is homotopic to f ′ (a homotopy h connecting f ′ and f ′′ can be deﬁned by h(y, t) = f ′ (y) + (M − m)t · x0 ) and has the property: f ′′ (y) > M for every y ∈ S k . Finally, letting y0 ∈ E0 ∩ K be any point with x∗ (y0 ) > M , we see that the straight-line homotopy connecting f ′′ and the constant map into y0 , lies in Un . Thus f is homotopically trivial, and Un is an AR for every n. By Ex.15 to §1.2, K∞ ∼ = Q. Exercises to §5.1. 1. Let X be a non-locally compact closed convex set. Show that X is not locally compact at each point x ∈ X. 2. Let X be a separable metrizable convex set of a topological vector space L and let E = span X. Then, there is a metrizable vector topology τ on E inducing on X its original topology. Hint: see T.Dobrowolski, H.Toru´ nczyk [1979]. A map f : X → Y between convex sets is called aﬃne if f (tx + (1 − t)x′ ) = tf (x) + (1 − t)f (x′ ) for t ∈ I and x, x′ ∈ X. An aﬃne map f : X → R is called an aﬃne functional. 3. Suppose X is a (closed) locally compact convex set. Show that every injective aﬃne map deﬁned on X is a (closed) embedding.

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4. Suppose X is a closed locally compact set in a linear metric space L. Show that X is ¯ of L. closed in the completion L 5. Show that a locally compact (closed) convex set X admits a (closed) aﬃne embedding into l2 if and only if X admits a point-separating sequence {x∗n } of aﬃne functionals. Hint to Ex.3–5: see C.Bessaga, T.Dobrowolski [199?]. A convex set aﬃnely homeomorphic to a convex compact subset of l2 is called a Keller cube. 6. Give an example of a convex compactum that is not a Keller cube. Hint: Consider Roberts compacta, see J.W.Roberts [1976], [1977]. 7. Show that every complete inﬁnite-dimensional convex set contains a Keller cube. Hint: see D.Curtis, T.Dobrowolski, J.Mogilski [1984].

§5.2. Topologically complete convex sets We start with studying SDAP in convex sets. ¯ is not locally 5.2.1. Proposition. Every convex set X whose completion X compact, satisﬁes SDAP. Proof. Suppose X is a convex subset of a linear metric space. Without loss ¯ is compact. Consider of generality, 0 ∈ X and no neighborhood of 0 in X the set Y = {(x, t) ∈ L × [0, ∞) | x ∈ t · X} ⊂ L ⊕ R and notice that Y is an additive subset in L ⊕ R and no neighborhood of 0 in Y is totally bounded. Then by Lemma 4.2.2, the space Y \{0} satisﬁes SDAP. Since Y \{0} is homeomorphic to X × R, by Ex.9 to §1.3, X has SDAP. This proposition and Characterization Theorem 1.1.14 imply 5.2.2. Corollary. Every non-locally compact complete convex set X ∈ AR is homeomorphic to s. This corollary together with Theorem 5.1.3 imply2 5.2.3. Corollary. Every complete convex set X ∈ AR is homeomorphic to [0, 1]n × (0, 1)m × [0, 1)p for some n, m ∈ N ∪ {0, ∞}, p ∈ {0, 1}. 5.2.4. Theorem. Every inﬁnite-dimensional Polish convex set X whose ¯ is an AR is homeomorphic to Q\Z for some Zσ -set Z ⊂ Q. completion X ¯ can be embedded into Q as a homotopy Proof. It follows from 5.2.3 that X ¯ ∈ AR, X is homotopy dense dense subset. Since X is a dense convex set in X ¯ in X, and thus X is a homotopy dense Gδ -subset in Q. Then X = Q\Z, where Z = Q\X is a Zσ -set in Q. Notice that every set of the form Q\Z, where Z is a Zσ -set in Q, is homeomorphic to a convex set, see Ex.2. 2 Banakh and Cauty [2010] generalized Corollary 5.2.3 to non-separable spaces and proved that each completely metrizable convex set X in a Fr´ echet space is homeomorphic to ℓ2 (κ) × [0, 1]m × [0, 1)p for some cardinals κ, m ≤ ω and p ≤ 1.

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¯ is an AR, 5.2.5. Proposition. Every convex set X whose completion X satisﬁes LCAP. ¯ satisﬁes LCAP. Since X is homotopy Proof. It follows from 5.2.3 that X ¯ X has LCAP accordingly to Ex.12d of §1.3. dense in X, ¯ to be not locally compact in 5.2.1 in too In fact, the condition on X restrictive, e.g., the completion s¯ = Q is compact; in the meantime s satisﬁes SDAP. Below we give some other conditions that imply SDAP for convex sets. By aﬀ X we denote the aﬃne hull of a subset X in a linear space. ¯ is an 5.2.6. Proposition. Suppose X is a convex set whose completion X ¯ AR. If X ̸⊂ aﬀ(X), then X has SDAP. ¯ Now we will show Proof. Obviously, X is a homotopy dense subset in X. ¯ Pick any point x0 ∈ X\ ¯ aﬀ(X) that X is also homotopy negligible in X. and notice that the set Y = {tx0 + (1 − t)x | t ∈ (0, 1], x ∈ X} lies in ¯ aﬀ(X), and thus X ∩ Y = ∅. Since Y is convex and dense in X, ¯ Y is X\ ¯ ¯ Since X ¯ homotopy dense. Then X ⊂ X\Y is homotopy negligible in X. has LCAP, we can apply Ex.12h of §1.3 to conclude that X has SDAP. 5.2.7. Corollary. Suppose X is a convex set closed in aﬀ(X) and such ¯ ∈ AR. If X is not locally compact, then X has SDAP. that X ¯ is not locally compact then X has SDAP by 5.2.1. So, we Proof. If X ¯ is locally compact, and thus X ̸= X. ¯ Since X is closed in suppose that X ¯ ∩ aﬀ(X), and thus X ¯ ̸⊂ aﬀ(X). Proposition 5.2.6 completes aﬀ(X), X = X the proof. 5.2.8. Corollary. Suppose X is a topologically complete convex set such ¯ is a compact AR and X ¯ ̸⊂ aﬀ(X) (this happens, e.g., X is a closed that X ¯ X) is homeomorphic to (Q, s). non-compact set). Then the pair (X, ∼ ¯ X\X) ¯ Proof. We will prove that (X, = (Q, Σ). For that, we use Theorem ⃗ 1.7.6 and the fact that the pair (Q, Σ) is C-absorbing, where C⃗ = {(K, K) | ¯ K ∈ M0 }. It follows from the proof of 5.2.6 that X\X is a homotopy dense ¯ Thus X\X ¯ ¯ such that (X\X, ¯ ¯ Zσ -subset in X. is a Zσ -subset in X X\X) ∈ ⃗ ¯ ¯ ⃗ σ C. Now to prove that (X, X\X) is strongly C-universal, it is enough to show that for every compactum K, every closed subset B ⊂ K, and every ¯ such that f |B is an embedding with f (B) ⊂ X\X, ¯ map f : K → X ¯ ¯ there is an embedding f : K → X\X close to f and coinciding with f ¯ we may approximate the map f on B. Since X is homotopy dense in X, ′ ′ by a map f such that f (K\B) ⊂ X and f ′ |B = f |B. By 5.2.7, X has SDAP, and thus, by 1.1.14, X is homeomorphic to s. Then, using strong M1 -universality of s, we may approximate the map f ′ |K\B by a closed ¯ deﬁned embedding g : K\B → X so close to f ′ that the map f ′′ : K → X

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by f ′′ |B = f |B, and f ′′ |K\B = g is continuous and injective. Fix any ¯ aﬀ(X) and let λ : K → I be a function with λ−1 (0) = B. Deﬁne x0 ∈ X\ ¯ by the formula ﬁnally for an ε > 0, the map f¯ : K → X f¯(x) = (1 − ελ(x))f ′′ (x) + ελ(x)x0 , x ∈ K. ¯ Evidently, f¯(K) ⊂ X\X. Moreover, taking ε suﬃciently small, we get f¯ is near to f . 5.2.9. Proposition. Suppose C is a topological closed-hereditary class of spaces such that C does not contains the Hilbert cube and has the following∪property: a space C belongs to C whenever C can be written as ∞ C = n=1 Cn , where Cn ∈ C for all n. Suppose X ∈ C is an inﬁnite¯ is an AR. Then X satisﬁes SDAP. dimensional convex set such that X Proof. Without loss of generality, 0 ∈ X. Fix any countable dense subset A = {an }∞ n=1 in X. Arguing as in Proposition 5.2.6, we see that it is enough ¯ such that the set to ﬁnd a point x0 ∈ X Y = {tx0 + (1 − t)a | t ∈ (0, 1], a ∈ conv A} does not intersect X. ¯ , where In fact, it suﬃces to ﬁnd a point x0 ∈ X\F F = { 1t x − 1−t t a | t ∈ (0, 1], x ∈ X, a ∈ conv A}. ∪∞ Remark that F ⊂ n=1 Fn , where Fn = {t0 x −

n ∑

ti ai | x ∈ X,

1 n

≤ t0 ≤ n, 0 ≤ ti ≤ n, i = 1, . . . , n}.

i=1

It is∑easy to see that for every n the set Fn is contained in the set Un = n −n i=1 ai + (2n + 1)n · X, which is aﬃnely homeomorphic to X, and hence, ∪belongs to the class C. Therefore, the set F is contained in the union U = n=1 Un . By the properties of C, U ∈ C. It follows from 5.2.3 that X contains a topological copy of the Hilbert cube Q. Since Q ∈ / C, Q ̸⊂ U , ¯ . and thus one can ﬁnd a point x0 ∈ Q\U ⊂ X\F 5.2.10. Corollary. Every countable-dimensional inﬁnite-dimensional con¯ ∈ AR satisﬁes SDAP. vex set X with X Finally, let us consider the question when a convex set is a Zσ -space. 5.2.11. Proposition. Suppose X ∈AR is a σ-compact inﬁnite-dimensional ¯ either is non-locally convex set that contains no Hilbert cube, and suppose X ¯ compact or X ∈ AR. Then X is a strong Zσ -space. Proof. It follows from 5.2.1 or 5.2.9 that X satisﬁes SDAP. By 1.4.9, each compactum in X is a strong Z-set. Since X is σ-compact, X is a strong Zσ -space.

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5.2.12. Proposition. Suppose X ∈ AR is closed convex set. If X ∈ M21 \M1 then X is a Zσ -space. This proposition results from the following lemma. 5.2.13. Lemma. Let D be a [0, 1]-stable local class of spaces such that a perfect image p(D) of any D ∈ D belongs to the class D. Assume that X∈ / D is a closed convex set. Then any closed set A ∈ D in X is a Z∞ -set. Proof. Let U ∈ cov(X) and a map f : I n → X of a ﬁnite-dimensional cube be given. Since X ∈ / D, there is a point x0 ∈ X that has no neighborhood belonging to the class D. By compactness of I n , there are ε > 0 and an open neighborhood U of x0 such that for every x ∈ U the map f¯x : I n → X deﬁned by f¯x = (1 − ε)f (t) + ε · x, t ∈ I n , is U-close to f . Notice that the map α : A×I n → L deﬁned by α(a, t) = ε−1 (a−(1−ε)f (t)), (a, t) ∈ A×I n , is perfect. Hence, α(A × I n ) ∈ D, and U ∩ α(A × I n ) ∈ D. Since U ∈ / D, there exists x ∈ U \α(A × I n ). Letting f¯(t) = (1 − ε)f (t) + ε · x, t ∈ I n , we see that (f¯, f ) ≺ U and f¯(I n ) ∩ A = ∅, i.e. A is a Z∞ -set in X. Exercises and Problems to §5.2.. 1. Let X be a closed convex set. a) Supposing that X is not locally compact, show that each compactum is a Z∞ -set in X; b) supposing that X is not complete, show that each complete subset in X is a Z∞ -set in X. 2. Show that every set of the form Q\Z, where Z is a Zσ -set in Q, is homeomorphic to a convex subset in l2 . Hint: In place of Q, consider the closed unit ball in l2 , equipped with the weak topology. 3. Give an example of a convex set X in l2 such that X is of the ﬁrst Baire category, but not a Zσ -space. Hint: see 5.5.19. 4. Give an example of a convex Borel set in l2 such that X is a Baire space but not a co-Zσ -space. 5. Give an example of a closed convex set X ∈ / M1 that is a co-Zσ -space. 6. (Open Problem) Does there exist a closed convex set such that X is a Borel Baire space, but not a co-Zσ -space? 7. Suppose X ∈AR is a convex set satisfying LCAP. Show that X satisﬁes SDAP, provided one of the following conditions is satisﬁed: ¯ ̸⊂ aﬀ(X); a) X b) X is closed and non-locally compact; c) X is countable-dimensional; d) X is σ-compact and contains no Hilbert cube; e) X ∈ C, where C is a class from 5.2.9. 8. Suppose A ⊂ Q is a closed subset that is not a Z-set in Q. Show that conv(A) contains a Hilbert cube. Hint: see T.Dobrowolski [1986a]. 9. (Open Problem) Does every convex set X ∈ AR satisfy LCAP?

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10. (Open Problem) Suppose X ∈ AR is a non-locally compact closed convex set. Does X satisfy SDAP?, LCAP?

§5.3. Strong universality in convex sets In this section we inspect the strongly C-universal property in convex sets. The main (unsolved) problem here is if every closed convex set X ∈ AR is strongly Z(X)-universal, where Z(X) stands for the class of all Z-sets in X. Remark that historically, Z-sets were introduced by R.D.Anderson in order to replace the non-topological conception of a subset of inﬁnite codimension. Let L be a linear space. Recall that a linear subspace L0 ⊂ L is said to have inﬁnite codimension in L, if the coset space L/L0 is algebraically inﬁnite-dimensional. Given two sets A ⊂ X in L we say that A has inﬁnite codimension in X, provided span A has inﬁnite codimension in span X. Recall that a subset X of a linear metric space L is called bounded if for every neighborhood U of 0 ∈ L, there exists n ∈ N with X ⊂ n · U . 5.3.1. Proposition. Suppose A is a subset of inﬁnite codimension in a convex set X ∈ AR. Then A is homotopy negligible in X. Proof. Supposing that X is a convex subset in a linear metric space L, denote by πA : L → L/ span A the coset map. Since span X/ span A is inﬁnitedimensional, we can choose inductively a dense in X sequence {xn } such that the vectors {πA (xn )} are linearly independent. Then B = conv{xn | n ∈ N} is a dense convex subset in X that misses A. By Ex.13 to §1.2, B is homotopy dense in X. Then, A ⊂ X\B is homotopy negligible in X. 5.3.2. Theorem. Let X be a convex set in a linear metric space L and A a closed bounded subset in L such that A ⊂ X and A has inﬁnite codimension in X. If X is an AR with LCAP, then X is strongly A-universal. Proof. To verify that X is strongly A-universal, ﬁx a cover U ∈ cov(X), a closed subset B ⊂ A, and a map f : A → X such that f |B : B → X is a Z-embedding. Since X is an AR with LCAP, every Z-set in X is strong. In particular, f (B) is a strong Z-set in X. Then according to 1.4.7, we may assume that f (C\B) ∩ f (B) = ∅ and f is closed over the set f (B). Let A′ = A\B, X ′ = X\f (B), and U ′ ∈ cov(X ′ ) be a cover such that St U ′ ≺ U and St U ′ ≺ {B(x, d(x, f (B))/2) | x ∈ X ′ }. By Ex.12c) to §1.3, the open subspace X ′ in X has LCAP. Thus, there is a map p : X ′ → X ′ such that (p, id) ≺ U ′ and the closure F = ClX ′ (p(X ′ )) is locally compact. By the choice of the cover U ′ , the map p¯ = p ∪ id : X ′ ∪ f (B) → X is continuous. Clearly, the map f ′ = p¯ ◦ f is U ′ -close to f .

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Since A has inﬁnite codimension in X, one can ﬁnd a countable dense subset S ⊂ X such that span(A) ∩ span(S) = {0} and span(S ∪ A) has inﬁnite codimension in X. Let x0 ∈ X\ span(S ∪ A) be any point. The set conv S, being convex and dense, is homotopy dense in X. Replacing p by a near map, if necessary, we may assume that F ⊂ conv S. Since F ⊂ X ′ is closed and locally compact, there is a locally ﬁnite cover W ∈ cov(X ′ ), W ≺ U ′ , such that for every W ∈ W the intersection W ∩ F is compact. Using the fact that A is bounded, construct a continuous function ε : X → I such that ε−1 (0) = f (B) and every x ∈ F has a neighborhood ε(x) W ∈ W such that (1 − ε(x))x + ε(x) 2 x0 + 2 A ⊂ W . Deﬁne a map f¯ : A → X by the formula (1)

ε ◦ f ′ (a) ε ◦ f ′ (a) f¯(a) = (1 − ε ◦ f ′ (a)) f ′ (a) + x0 + · a. 2 2

We claim that f¯ : A → X is a Z-embedding with f¯|B = f |B and ¯ (f , f ) ≺ U. It follows from the construction that the last two conditions are satisﬁed. Since f¯ is closed over f (B), and f¯(A\B) ∩ f (B) = ∅, to show that f¯ is a Z-embedding, it suﬃces to prove that f¯|A\B → X\f (B) is a Z-embedding (see Ex.5 to §1.4). By the choice of x0 and S, the equality f¯(a) = f¯(a′ ), where a, a′ ∈ A\B, implies ε ◦ f ′ (a) = ε ◦ f ′ (a′ ) and a = a′ . Thus, the map f ′ is injective. To show that f¯|A\B : A\B → X\f (B) is a closed embedding, it now suﬃces to verify that this map is perfect. Fix a compactum K ⊂ X\f (B). We have to show that the preimage K − = f¯−1 (K) ⊂ A\B is compact. Since f¯ is closed over f (B) the set K − is closed not only in A\B but also in A. Since (f¯, f ′ ) ≺ W, we have f ′ (K − ) ⊂ St (K, W). By the choice of the cover W, the set M = ClX (St (K, W)) ∩ F is compact. Let ε0 = min{ε(x) | x ∈ M } and remark that the set D = [1, ε20 ](K − [0, 1]M − [0, 1]x0 ) ⊂ L is compact. It follows from (1) that for every a ∈ K − ( ) 2 ε ◦ f ′ (a) ′ ′ ¯ a= f (a) − (1 − ε ◦ f (a)) f (a) − x0 ∈ D. ε ◦ f ′ (a) 2 Thus K − ⊂ D ∩ A is compact and f¯ : A → X is a closed embedding. To see that f¯(A) is a Z-set in X, notice that f¯(A) ⊂ f (B) ∪ conv(A ∪ {x0 } ∪ S) and the set conv(A ∪ {x0 } ∪ S), having inﬁnite codimension, is homotopy negligible in X. 5.3.3. Corollary. Let X be a closed convex non-locally compact subset in a normed space L. If X ∼ = X × Y for some inﬁnite-dimensional convex set Y then X is strongly universal.

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¯ ∈ AR. Then by 5.2.7, X satisﬁes SDAP, and conseProof. Clearly, X ∼ quently, X × Y = X satisﬁes LCAP, see Ex.12e to §1.3. Let ∥ · ∥ denotes the norm of L and Xn = {x ∈ X : ∥x∥ ≤ n}. Clearly, each Xn is a closed bounded set of inﬁnite codimension in X × Y . Then by 5.3.2, X × Y ∼ =X is strongly Xn -universal for every n, i.e. Xn ∈ SU (X). Since the class SU (X) is local (see 1.5.8), X ∈ SU (X). 5.3.4. Corollary. Suppose X is a closed convex non-locally compact subset of a normed space such that X ∼ = X×Y for some inﬁnite-dimensional convex set Y . The space X is (co)absorbing if and only if X is a (co-)Zσ -space. 5.3.5. Proposition. Let C be a 2ω -stable M1 -hereditary weakly A1 -additive compactiﬁcation-admitting class of spaces, and let X be a convex set in a linear metric space L such that X is an AR with SDAP. Then X is strongly C-universal, provided X contains a C-universal subset D, closed in L. ¯ ∩L = Proof. Suppose D is a C-universal subset of X, closed in L. Then D ¯ ∩ X = D. By Theorem 3.1.1, the pair (D, ¯ D) is (M0 ∩ C, C)-universal. D According to 3.2.18, to show that X is strongly C-universal, it suﬃces to show that so is the product X × Q. By Ex.8 and 12 to §1.3, X × Q satisﬁes LCAP. Fix any C ∈ C and ﬁnd a compactum K ∈ C containing C. ¯ D) = (D, ¯ D ¯ ∩ X), ﬁnd an embedding Using (M0 ∩ C, C)-universality of (D, −1 ¯ e : K → D with e (X) = C. Fix any point q ∈ Q. Identifying X with X × {q} ⊂ X × Q, we get C is homeomorphic to a closed bounded subset e(C) of inﬁnite codimension in X × Q. By Theorem 5.3.2, the space X × Q is strongly C-universal. 5.3.6. Theorem. Suppose Ω is a C-absorbing AR, where C satisﬁes the hypotheses of 5.3.5. A closed convex set X is homeomorphic to Ω if and only if (1) X ∈ AR is a Zσ -space satisfying SDAP; (2) X ∈ σC; (3) X is C-universal. 5.3.7. Theorem. Suppose Ω is a C-coabsorbing AR, where C satisﬁes the hypotheses of 5.3.5. A closed convex set X is homeomorphic to Ω if and only if (1) X ∈ AR is a co-Zσ -space satisfying SDAP; (2) every Z-set of X belongs to the class C; (3) X is C-universal. 5.3.8. Proposition. Let 0 ≤ n ≤ ∞ and X a convex set in a linear metric space L such that X is an AR with LCAP. Then X is strongly M1 [n]-universal, provided X contains an M1 [n]-universal subset D, closed in L.

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157

Proof. Suppose D ⊂ X is a closed in L M1 [n]-universal subset. To prove that X is strongly M1 [n]-universal, we will apply Theorem 3.2.19. Fix an open set U ⊂ X, a cover U ∈ cov(U ), and a map f : C → U , where C ∈ M1 [n]. Let V ∈ cov(U ) be a cover with St V ≺ U . By Ex.12c from §1.3, U satisﬁes LCAP. Using this fact, construct a map f ′ : C → U such that K = ClU (f ′ (C)) is locally compact and (1)

(f ′ , f ) ≺ V.

Pick a locally ﬁnite cover W ∈ cov(U ) such that W ≺ V and for every W ∈ W the intersection W ∩ K is compact. Let M = l2 if n = ∞, and let M be the n-dimensional N¨obeling space if n < ∞. Since the space D is M1 [n]-universal, there is a closed embedding M ⊂ D. Let x0 ∈ M be any point. By continuity of linear operation, there exists a continuous function ε : K → (0, 1] such that for every x ∈ K there is a neighborhood W ∈ W of x such that for every y ∈ O(x0 , 2ε(x)) we have (2)

(1 − ε(x))x + ε(x)y ∈ W.

∪k 1 For every k ∈ N let Ck = (ε ◦ f ′ )−1 ([ k+1 , k1 ]), and C˜k = i=1 Ci . Repeating the arguments from 4.2.5, construct a closed embedding e : C → M \{x0 } such that for every k ∈ N we have 0 < inf{d(e(c), x0 ) | c ∈ Ck } ≤ sup{d(e(c), x0 ) | c ∈ Ck }

0, there exists a map g : A → Xm for some m ≥ n, such that g|B = f |B and d(f, g) < ε. Proof. We will construct maps f0 , f1 : A → Xm for some m ≥ n, such that f0 |B = f |B and d(f1 , f ) < ε/2. Then for any Urysohn map λ : A → [0, 1] such that λ(B) = {0} and {a ∈ A : |f0 (a) − f (a)| ≥ ε/2} ⊂ λ−1 (1), the required map g may be deﬁned by the formula g(a) = (1 − λ(a))f0 (a) + λ(a)f1 (a). The map f0 is obtained as an extension of the map f |B into the AR-space Xn . In constructing the map f1 , we may assume that A is a Hilbert cube, since X is an AR. For every k we identify the k-dimensional cube I k with the subset {(ti ) ∈ Q | ti = 0 for i > n} of Q. Denoting by prk : Q → I k the projection onto the ﬁrst k-coordinates, ﬁnd k such that the map f ◦ prk ε is 4ε -near to f . Let N be a triangulation of I k with diam f (σ) < 8(k+1) ∪ ∞ for σ ∈ N . For every vertex v of N , let f˜(v) ∈ i=1 Xi be any point ε such that d(f˜(v), f (v)) < 8 . Deﬁne ﬁnally the map f˜ : I k → X letting ∑k ∑k f˜(q) = i=0 ti f˜(vi ) for q = i=0 ti vi a point of a closed k-simplex in N with vertices v0 , . . . , vk . Similarly as in 5.1.1, show that the map f˜ is 4ε -close to f |I k . Clearly, f˜(I k ) ⊂ conv f˜(N (0) ) ⊂ Xm for some m ≥ n. Then the map f1 = f˜ ◦ prk is as required. Proof of 5.3.10. Suppose X ∈ AR is an inﬁnite-dimensional convex set. Pick a countable dense set {xi } in X. Because of inﬁnite-dimensionality of X, we may assume that the vectors x1 , x2 , . . . are linearly independent. For

5.3. STRONG UNIVERSALITY IN CONVEX SETS

159

every n put Xn = conv{x1 , . . . , xn }. We claim that the tower X1 ⊂ X2 ⊂ . . . is an M0 (ω)-skeleton in X. This together with Ex.5 to §1.5 will imply that X is strongly M0 (ω)-universal. Firstly, notice that each Xn is a ﬁnite-dimensional compact Z-set in X (the last can be proved by the technique from 5.1.1). Fix ε > 0, a ﬁnite-dimensional compactum A, a closed subset B ⊂ A, and a map f : A → X such that f (B) ⊂ Xn for some n. By Lemma 5.3.11, this map can be approximated by a map g : A → Xm for some m ≥ n, so that g|B = f |B and d(f, g) < 2ε . Evidently, the quotient space A/B is ﬁnite-dimensional. Thus, there is an embedding e : A/B → I k for some k such that e({B}) = (0, . . . , 0). Denote by π : A → A/B the quotient map and for a ∈ A let e(a)i denote the i-th coordinate of the point e(a) ∈ I k . Finally, deﬁne a map f¯ : A → Xm+k by the formula k k ( ) ∑ ∑ f¯(a) = 1 − δ e(a)i g(a) + δ e(a)i xm+i , i=1

i=1

where δ < 1/k is so small that the map f¯ is to g. It is easy to verify ¯ that the map f is injective and has the properties: d(f¯, f ) < ε, f¯|B = f |B, and f¯(A) ⊂ Xm+k . ε 2 -near

5.3.12. Corollary. Let X ∈ A1 (s.c.d.) be an inﬁnite-dimensional convex ¯ of X is either non-locally compact or X ¯ ∈ set. Suppose the completion X AR. Then X is homeomorphic to σ. Proof. By Haver Theorem 1.1.7, X is an AR, by 5.2.11, X is a strong Zσ space, and by 5.3.10, X is strongly M0 (ω)-universal. Since X ∈ A1 (s.c.d.) = σM0 (ω), we get that X is an M0 (ω)-absorbing AR. By the same reason, σ is an M0 (ω)-absorbing AR. Then by Theorem 1.6.3, X is homeomorphic to σ. Now, let us consider the question when a convex set in strongly M0 universal. Recall that a convex set aﬃnely homeomorphic to a convex compact subset of l2 , is called a Keller cube. It follows from 5.1.2 that each Keller cube is homeomorphic to Q. 5.3.13. Theorem. Every convex set X ∈ AR containing a Keller cube is strongly M0 -universal. For the proof we need an auxiliary lemma that concerns central points. We deﬁne a point x0 of a Keller cube K to be central if for every t ∈ [0, 1) (1 − t)x0 + t · K is a Z-set in K (for more details on central points, see C.Bessaga, A.Pelczy´ nski [1975, V.§4]).

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5.3.14. Lemma. Let K be a Keller cube in a linear metric space E. Then for every ﬁnite set {x1 , . . . , xn } in E the set L = conv{K, x1 , . . . , xn } is also a Keller cube. Furthermore, there exists z ∈ K such that z is a central point for every such L. Proof. Let α : K → l2 be an aﬃne embedding. We may assume that 0 ∈ K and α(0) = 0 ∈ l2 . For any L = conv{K, x1 , . . . , xn } the map α can ∪∞ be extended to an aﬃne embedding of L as follows. If x1 ∈ span K = n=1 n(K − K), say x1 = n(k1 − k2 ), set α(x1 ) = n(α(k1 ) − α(k2 )). And if x1 ∈ / span K, choose α(x1 ) ∈ l2 \ span α(K). Then α extends linearly to a homeomorphism between conv{K, x1 } and conv{α(K), α(x1 )}. Repeating this procedure n times, we obtain the desired extension of α over L, with α(L) = conv{α(K), α(x1 ), . . . , α(xn )}. By the foregoing, we may assume without loss of generality that E = l2 and 0 ∈ K. Choose an orthogonal sequence {ui } of nonzero vectors in the inﬁnite-dimensional pre-Hilbert space span K. We may suppose that each ui ∈ K − K; pick vi , wi ∈ K such that ui = ∑ vi − wi . Since K is compact, ∞ the sequence {wi } is bounded. Consider z = i=1 2−i wi . We have z ∈ K, −i and z + 2 ui ∈ K for each i. Then for every convex compact set L with K ⊂ L ⊂ l2 , we have inf{|c| : z + cun ∈ / L} > 0 for each n. Then by Proposition 4.3 from C.Bessaga, A.Pelczy´ nski [1975, V.§4], z is a central point of Z. Proof of 5.3.13. Suppose K ⊂ X is a Keller cube. Let z ∈ K be a point with the property speciﬁed in 5.3.14. We may assume z = 0. Let {xi } be a dense sequence in X, and let Ln = conv{K, x1 , . . . , xn } for n ∈ N. By the choice of z = 0, for every t ∈ [0, 1) and every n, the Keller cube t · Xn is a Z-set in Xn . Let {tn } be a strictly increasing sequence of positive numbers such that tn → 1. For every n let X∪ n = tn · Ln . Then {Xn } is a tower of ∞ convex sets, with each Xn ∼ = Q and n=1 Xn is dense in X. Since the pair (tn+1 Ln+1 , tn Ln+1 ) is homeomorphic to the pair (Ln+1 , tLn+1 ) for some 0 < t < 1, tn Zn+1 is a Z-set in tn+1 Ln+1 . By Z-Set Unknotting Theorem 1.1.25, the pair (Xn+1 , Xn ) is homeomorphic to (Q × Q, Q × {pt}). We claim that X1 ⊂ X2 ⊂ . . . is an M0 -skeleton in X. First, notice that each Xn is a Z-set in X. Indeed, given a map f : I n → X, we may approximate f by a map f ′ : I k → Xm for some m > n (just apply Lemma 5.3.11), and then use the fact that Xn is a Z-set in Xm . Fix ε > 0, a map f : A → X of a compactum A, and a closed subset B ⊂ A such that f |B is injective and f (B) ⊂ Xn for some n. By Lemma

5.3. STRONG UNIVERSALITY IN CONVEX SETS

161

5.3.11, the map f can be approximated by a map f ′ : A → X such that d(f, f ′ ) < 2ε , f ′ |B = f |B, and f ′ (A) ⊂ Xm for some m > n. Clearly, g|B : B → Xm is a Z-embedding. Then using the strong M0 -universality of Q ∼ = Xm (see 1.1.26), we can ﬁnd a Z-embedding f¯ : A → Xm such that ¯ f |B = f ′ |B = f |B and d(f¯, f ′ ) < 2ε (and thus d(f¯, f ) < ε). Therefore, the tower {Xn } is an M0 -skeleton in X. Then by Ex.5 to §1.5, the space X is strongly M0 -universal. 5.3.15. Corollary. Suppose X ∈ AR is a σ-compact convex set satisfying SDAP. If X contains a Keller cube then X is homeomorphic to Σ. Now let us return to Theorem 5.3.2. One can see that the assumption on A to be bounded in this theorem is essential. In the meantime, Corollary 5.3.3 can be generalized. 5.3.16. Theorem. Let X ∈ AR be a closed convex set in a linear metric space and Y ∈ AR a convex set satisfying SDAP. Then the space X × Y is strongly X-universal. Proof. Suppose X, Y are convex sets in linear metric spaces LX , LY respectively. Without loss of generality, 0 ∈ X and 0 ∈ Y . We will identify X × Y with the subspace X + Y in LX ⊕ LY . Since Y has SDAP, the product X × Y has SDAP either, see Ex.8 to §1.3. Then by 1.5.7, to show that X + Y is strongly X-universal, it suﬃces for every open sets V ⊂ X, U ⊂ X + Y , every cover U ∈ cov(U ), and every map f : V → U to ﬁnd a Z-embedding f¯ : V → U , U-close to f . Clearly, the open subspace U of X +Y has SDAP as well as LCAP. Then, replacing f by a near map, if necessary, we may assume that the closure F = ClU (f (V )) is locally compact. Since X has inﬁnite codimension in X + Y , one can select inductively a countable dense subset S ⊂ X + Y such that span(S) ∩ LX = {0}, S ∩ Y is dense in Y , and span(X ∪ S) has inﬁnite codimension in X + Y . Let x0 ∈ (X + Y )\ span(X ∪ S) be any point. Clearly, the set conv(S) is homotopy dense in X + Y and thus replacing f by a near map, if necessary, we may suppose that F ⊂ conv(S). Pick a locally ﬁnite cover W ∈ cov(U ) ¯ ∩ F is compact. such that W ≺ U and for every W ∈ W the intersection W Let ε : U → (0, 1] be a function such that {B(x, 4ε(x))}x∈U ≺ W. Using continuity of the linear operations, construct a map ε1 : U → (0, 1] so that ε1 (x) ≤ ε(x), ∥ − ε1 (x) · x∥ ≤ ε(x), and ∥ε1 (x) · x0 ∥ ≤ ε(x) for x ∈ U . Next, construct a map δ : X → [0, 1] such that δ −1 ((0, 1]) = V , δ(x) ≤ 1 3 ε1 ◦ f (x), and ∥δ(x) · x∥ ≤ ε ◦ f (x) for x ∈ V . Since Y is an AR with SDAP, it is inﬁnite-dimensional, and by Theorem 5.3.10, I 2 ∈ SU (Y ). Since Y has SDAP, the class SU (Y ) is open-hereditary, see 1.5.8, and thus (0, 1]2 ∈ SU (Y ). By 1.5.6, the open subspace Y \{0} of Y is strongly (0, 1]2 -universal.

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It is easy to construct an embedding i : I → Y such that t 3 ≤ ∥i(t)∥ ≤ t for t ∈ I. 2 4

(1)

Using the strong (0, 1]2 -universality of Y \{0}, ﬁnd a perfect map p : (0, 1]2 → Y \{0} such that ∥p(t, t′ ) − i(t)∥ ≤

(2)

1 ∥i(t)∥ 4

for (t, t′ ) ∈ (0, 1]2 .

By the choice of S, S ∩ Y is dense in Y , and thus Y ∩ conv S is homotopy dense in Y . Replacing p by a close map, if necessary, we may assume that p((0, 1]2 ) ⊂ conv S. By (1) and (2), 1 t ≤ ∥p(t, t′ )∥ ≤ t for (t, t′ ) ∈ (0, 1]2 . 4 Deﬁne ﬁnally the map f¯ : V → X + Y be the formula

(3)

ε1 ◦ f (x) (4) f¯(x) = (1 − ε1 ◦ f (x)) · f (x) + x0 + 3 ε1 ◦ f (x) + δ(x) · x + p(ε1 ◦ f (x), δ(x)). 3 We claim that f¯ is a Z-embedding U-close to f . Indeed, for an x ∈ V we have ∥f¯(x) − f (x)∥ = ∥ − ε1 ◦ f (x) · f (x)+ ε1 ◦ f (x) ε1 ◦ f (x) x0 + δ(x) · x + p(ε1 ◦ f (x), δ(x))∥ ≤ 3 3 ≤ ∥ − ε1 ◦ f (x) · f (x)∥ + ∥ε1 ◦ f (x) · x0 ∥ + ∥δ(x) · x∥ + ∥p(ε1 ◦ f (x), δ(x))∥ ≤ +

≤ ε ◦ f (x) + ε ◦ f (x) + ε ◦ f (x) + ε1 ◦ f (x) ≤ 4ε ◦ f (x). By the choice of ε, this implies f¯(V ) ⊂ U and (f¯, f ) ≺ W ≺ U. Similarly as in Theorem 5.3.2, one can verify that f¯ is injective and f¯(V ) is homotopy negligible in U . Let us show that f¯ is a perfect map. For, ﬁx a compactum K ⊂ U . We have to show that K − = f¯−1 (K) is compact. Notice that (f¯, f ) ≺ W implies f (K − ) ⊂ St (K, W). By the choice of W, the set M = ClU (St (K, W) ∩ F ) is compact. Then ε0 = min{ε1 (x) | x ∈ M } > 0. Let us consider the projection prY : X + Y → Y and notice that ε1 ◦ f (x) prY ◦ f¯(x) = (1 − ε1 ◦ f (x)) · prY ◦ f (x) + prY (x0 )+ 3 ε1 ◦ f (x) + p(ε1 ◦ f (x), δ(x)) 3

5.3. STRONG UNIVERSALITY IN CONVEX SETS

163

for x ∈ V . Then for any x ∈ K − we have p(ε1 ◦ f (x), δ(x)) ∈ D, where D = [1, ε30 ](prY (K) − [0, 1]prY (M ) − [0, 1]prY (x0 )). Since D is compact and p|[ε0 , 1] × (0, 1] is a perfect map, the set p−1 (D) ∩ [ε0 , 1] × (0, 1] is compact, and thus δ0 = min{δ(x) | x ∈ K − } > 0. Since δ −1 (0, 1] = V , K − is closed not only in V , but also in X. Let N = [1, δ0−1 ](K − [0, 1] M − [0, 1] x0 − [0, 1] D) and notice that by (4), every x ∈ K − belongs to N . Since N is compact and X is closed in LX ⊕ LY , the set K − ⊂ X ∩ N , being closed in X, is compact either. 5.3.17. Corollary. Suppose X ∈ AR is a closed convex set such that X ∼ = X ×Y for some convex set Y satisfying SDAP. The space X is (co)absorbing if and only if X is a (co-)Zσ -space. Finally, let us consider the question when a convex set is homeomorphic to a convex set in l2 . We pose this question a little bit more generally and will ask when a C-absorbing AR is homeomorphic to a convex set in l2 . We will need the following statement. 5.3.18. Proposition. Let X ∈ AR be an inﬁnite-dimensional convex set satisfying LCAP in a linear metric space L, and A ⊂ X be a bounded closed in L linearly independent subset. Then X is strongly A-universal. Proof. If A contains at most one limit point, then A is a ﬁnite-dimensional compactum, and X is strongly A-universal by 5.3.10. So, assume that A contains at least two limit points. In this case, we can write A = A1 ∪ A2 , where A1 , A2 are closed subsets in A with inﬁnite complements. Since A is linearly independent, we get both A1 and A2 have inﬁnite codimension in X. Then by Theorem 5.3.2, A1 , A2 ∈ SU (X). Since X satisﬁes LCAP, every Z-set in X is strong, see Ex.22 to §1.4. Then by 1.5.8, the class SU (X) is closed-additive and thus A ∈ SU (X). We deﬁne a class of spaces C to be multiplicative, provided A × B ∈ C for every A, B ∈ C. Recall that a linear metric space is called a pre-Hilbert space, provided it is linearly homeomorphic to a linear subspace in l2 . 5.3.19. Theorem. Suppose Ω is a C-absorbing AR, where C is a topological closed-hereditary multiplicative class of spaces. Then Ω is homeomorphic to a pre-Hilbert space. Proof. It is well known that l2 contains a linearly independent compact subset homeomorphic to Q, see C.Bessaga, A.Pelczy´ nski [1975, VIII,§2]. So we may consider Ω to be a dense subset of a linearly independent compact subset K in l2 . Clearly, span Ω is a pre-Hilbert space. Let us show that span Ω is homeomorphic to Ω. For this, according to Theorem 1.6.3, it suﬃces to verify that span Ω is a C-absorbing AR. First notice that Ω =

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APPLICATIONS II: CONVEX SETS

K ∩span Ω is a closed bounded linearly independent subset in span Ω. Thus, by 5.3.18, span Ω is strongly Ω-universal. Since the space Ω is C-universal, we get that span Ω is strongly C-universal. Since every closed inﬁnite-dimensional linear subspace of l2 is linearly homeomorphic to l2 , we may assume that span Ω is dense in l2 , and consequently, span Ω is homotopy dense in l2 . Since span Ω is contained in a σ-compact ( and thus Zσ -) set span K ⊂ l2 , homotopical density of span Ω implies that span Ω is a Zσ -space. By 5.2.1, the space span Ω satisﬁes SDAP. Now we are going to show that span Ω ∈ σC. Notice ﬁrstly that since Ω ∈ AR, Ω ∈ σC contains a topological copy of I. Using this fact by Baire Category argument, we can deduce that I ∈ C. It is easy to see that span K ∞ ∪ can be written as span K = {0} ∪ Kn,m , where n,m=1

Kn,m =

n {∑ i=1

ti xi |

1 1 ≤ |ti | ≤ m, xi ∈ K and d(xi , xj ) ≥ for i ̸= j}. m m

Then span Ω = {0} ∪

∞ ∪

Kn,m ∩ span Ω.

n,m=1

It is easy to see that the intersection Kn,m ∩ span Ω is homeomorphic to the closed subset {(t1 , . . . , tn , x1 , . . . , xn ) | 1 1 | ≤ |ti | ≤ m, xi ∈ Ω and d(xi , xj ) ≥ for i ̸= j} m m in [−m, m]n × Ωn . Since [−m, m] ∈ C, Ω ∈ σC, and the class C is closedhereditary topological and multiplicative, [−m, m]n × Ωn ∈ σC, and consequently, Kn,m ∩ span Ω ∈ σC. Since each Kn,m ∩ span Ω is closed in span Ω, we conclude span Ω ∈ σC. Thus span Ω is a C-absorbing AR, homeomorphic to Ω. 5.3.20. Corollary. Let X ∈ AR be a closed convex set satisfying SDAP. If X is a Zσ -space homeomorphic to its own square X × X, then X is an absorbing space homeomorphic to a pre-Hilbert space. Proof. By 5.3.17, the space X is F0 (X)-absorbing. Since X ∼ = X × X, the class F0 (X) is multiplicative, and by 5.3.19, the F0 (X)-absorbing space X is homeomorphic to a pre-Hilbert space.

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165

Exercises and Problems to §5.3. 1. (Open Problem) Let X ∈ AR be an M0 -universal convex set. Is then X strongly M0 -universal? The answer is not known even for closed convex subsets in pre-Hilbert spaces. 2. (Open Problem) Is every closed convex set X ∈ AR strongly Z(X)-universal? 3. (Open Problem) Suppose X ∈ AR is a closed convex set such that X is a (co-)Zσ space. Is X (co)absorbing? 4. Let K be a linearly independent compact subset in l2 . Show that span K contains no Keller cube. 5. Find an example of a convex set X ∈ A1 (s.c.d.) in l2 such that span X contains a Keller cube. Hint: Embed I as a linearly independent subset of l2 × {0} in l2 × l2 and let Φ : I → {0}×l2 ⊂ l2 ×l2 be a map onto a Keller cube. Show that conv(graph(Φ)∪I ×{0}) is as required. 6. Suppose X ∈ AR is a closed convex set such that X is a Zσ -space. Show that the following conditions are equivalent: a) X ∼ = X × Y for some convex set Y with SDAP; b) X ∼ = X × σ. 7. Show that the Hilbert cube Q contains a linearly independent compact subset homeomorphic to Q.

8. Applying Ex.6 and Ex.7 to §5.1, show that every complete linear metric space contains a linearly independent compact subset homeomorphic to Q. 9. Suppose Ω is a C-absorbing AR, where C is a topological closed-hereditary multiplicative class of spaces. Show that every complete linear metric space L ∈ AR contains a linear subspace homeomorphic to Ω. 10. Show that a closed convex set A of a linear metric space is a Z∞ -set in L if and only if either A has inﬁnite codimension in L or A is nowhere dense in aﬀ(A). Hint: see Banakh [1994]. 11. Show that for a space C, a linear metric AR-space X is strongly C-universal if and only if X contains a closed convex subset A such that A is a strongly C-universal AR. Hint: see Banakh [1998b]. 12. (Operator images) Suppose T : E → F is an injective linear continuous map between inﬁnite-dimensional Fr´ echet spaces such that T E is dense in F . a) Supposing T E ∈ M21 show that the spaces s, Σ, Σ × s exhaust all possible topological types of the space T (E) and the pairs (s, s), (s, Σ), (s × s, Σ × s), (s × Q, Σ × s), and (s × Q × s, Σ × s × s) exhaust all possible topological types of the pair (F, T E). b) Suppose T E ∈ M2 , E is not normable, and T is a bounded operator (that is T U is bounded for some open U ⊂ E). Show that T E ∼ = Ω2 . c) Suppose T E ∈ M2 , T is a compact operator, and the space E endowed with the weak topology is an M2 -universal space. Show that T E ∼ = Ω2 and (F, T E) ∼ = (Q × s, Ω2 × σ). d) Suppose T E ∈ M2 , E contains an isomorphic copy Z of the Banach space c0 , and T |Z is not an embedding. Show that T E ∼ = Ω2 . Hint: See T.Banakh, T.Dobrowolski, A.Plichko [2000].

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§5.4. Strong universality in locally convex spaces In this section under a locally convex space we understand an inﬁnitedimensional locally convex linear separable metric space. The main result of this section is ¯ denotes its 5.4.1. Theorem. Suppose E is a locally convex space and E ¯ ¯ completion. For every compactum A ⊂ E the pair (E, E) is strongly (A, A ∩ E)-universal. We postpone the proof of this theorem to the end of this section. Now let us consider some of its corollaries. First notice that combining 5.4.1 and 1.7.9 we obtain 5.4.2. Theorem. Let E be a locally convex space. For every closed totally bounded subset A ⊂ E the space E is strongly A-universal. We deﬁne a locally convex space E to be σ-precompact if E can be expressed as a countable union of its totally bounded subsets. Clearly, E is ¯ σ-precompact if and only if E lies in a σ-compact set in the completion E of E. 5.4.3. Theorem. Every σ-precompact locally convex space E is absorbing. Proof. We claim that E is Ftb (E)-absorbing, where Ftb (E) stands for the class of spaces homeomorphic to closed totally bounded subspaces of E. ¯ Write Since∪E is σ-precompact, E ⊂ A for some σ-compact set A ⊂ E. ∞ ¯ Because A = n=1 An , where each An is compact, and thus a Z-set in E. ¯ we get that E is a Zσ -space. By 5.2.1, of homotopical density of E in E, E satisﬁes SDAP, and by 5.4.2, E is strongly Ftb (E)-universal. Clearly E ∈ σFtb (E), and thus E is Ftb (E)-absorbing. Ex.1 from §1.6 completes the proof. 5.4.4. Theorem. Every σ-precompact locally convex space E is homeomorphic to a pre-Hilbert space. ¯ containing E. Write A = ∪∞ An , Proof. Let A be a σ-compact set in E n=1 where each An ⊂ An+1 is compact. Let {x∗n }∞ n=1 be a point-separating se¯ Without loss of generality, we quence of continuous linear functionals on E. may assume that supa∈An |x∗n (a)| ≤ n1 . Then the formula f (a) = (x∗n (a))∞ n=1 determines an injective map f : A → l2 . We claim that E ′ = f (E) is the required pre-Hilbert space, homeomorphic to E. Clearly, E ′ is a linear subspace in l2 . Because of compactness, for each n the restriction f |An is an ∪∞ embedding. Thus E ′ ⊂ n=1 f (An ) is a σ-precompact space. Denote by A the class of spaces homeomorphic to one of the spaces An ∩ E, n ∈ N. It follows from 5.4.3 and the construction of E ′ that both E and E ′ are A-absorbing AR’s. By 1.6.3, E and E ′ are homeomorphic.

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It should be noticed that Theorem 5.4.4 is false beyond the class of σprecompact locally convex spaces, see counterexamples 5.5.8 and 5.5.9. Applying Theorem 5.4.1, we can, in the case of locally convex spaces, improve Proposition 5.3.5. 5.4.5. Proposition. Let C be a 2ω -stable weakly A1 -additive compactiﬁcation-admitting class of spaces. A locally convex space E is strongly Cuniversal if and only if E is C-universal. ¯ E) is (M0 ∩ Proof. Suppose E is C-universal. Then by 3.1.1, the pair (E, ¯ C, C)-universal. According to Theorem 5.4.1, the pair (E, E) is strongly (M0 ∩ C, C)-universal. Finally applying 1.7.9, we get that the space E is strongly C-universal. Now we come to the proof of Theorem 5.4.1. It is well known that each locally convex space E admits an F -norm, i.e. a function ∥ · ∥ : E → [0, ∞) such that (1) the formula d(x, y) = ∥x − y∥ determines a compatible metric on E; (2) ∥t · x∥ ≤ ∥x∥ for t ∈ [−1, 1], x ∈ E; (3) the balls N (η) = {x ∈ E : ∥x∥ < η} are convex for all η > 0. Let E be a locally convex space and ∥ · ∥ an F -norm on E. We denote by ¯ the completion of E, by H(E) the set of all self-homeomorphisms of E, E ¯ E) the set of all homeomorphisms h : E ¯→E ¯ with h(E) = E. and by H(E, The proof of Theorem 5.4.1 depends on the following lemma. ¯ 5.4.6. Lemma. Let E be a locally convex space, ∥ · ∥ an F -norm on E, ¯ ¯ and let B and A be disjoint compacta of E. Then, given a map f : B → E ¯ and ε > 0, there exists a homeomorphism h ∈ H(E, E) satisfying (i) ∥h − id∥ ≤ ∥f − id|B∥ + ε; (ii) h|A = id|A; (iii) ∥h|B − f ∥ < ε. In the proof of 5.4.6 we use two lemmas. 5.4.7. Lemma. Assuming that span(B ∪ f (B)) ⊂ E is ﬁnite-dimensional ¯ E) and that f is an embedding with f (B) ∩ A = ∅, there exists h ∈ H(E, extending f and satisfying the conditions (i) and (ii) of 5.4.6. 5.4.8. Lemma. Given a positive δ, there exists a map ¯ u : E × [0, 1) ∪ B × {1} → E such that, writing ut = u(·, t), we have u1 (B) ∩ A = ∅, span(u1 (B)) ⊂ E is ﬁnite-dimensional and, for t < 1 ¯ E) with u0 = id; (iv) ut ∈ H(E, (v) ∥ut − id∥ < δ;

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(vi) ut |A = id|A. In the proof of 5.4.7 we use the following lemma. For its proof use ﬁrst the “graph-trick” of V.Klee and the technique of pseudotranslations, see C.Bessaga, A.Pelczy´ nski [1975, p.62 and IV.§6], to show that every embedding f : K → E of a compact subset K of a ﬁnite-dimensional linear space can be extended to a homeomorphism of E, provided dim E ≥ 2 dim span(K∪f (K)), and then repeat the arguments from §7 of T.Chapman [1976]. 5.4.9. Lemma. Let F be a ﬁnite-dimensional linear metric space, U an open cover of F , F0 a linear subspace of F , U and open set in F , K ⊂ U ∩ F0 a compact subset, and h : K × I → U ∩ F0 a U-homotopy such that h0 = id|K and h1 is an embedding. If dim F ≥ 4 dim F0 +4 then there exists a homeomorphism H : F → F such that H|K = h1 , H|F \U = id|F \U , and H is U-homotopic to id. Proof of 5.4.7. Pick a positive η with η < dist(A, B ∪ f (B)) and η < ε. The compactness of A implies that there is a ﬁnite-dimensional linear subspace E ′ of E satisfying (1)

A ⊂ E ′ + N (η/16)

and such that E1 = span(B ∪ f (B)) ⊂ E ′ and dim(E ′ ) ≥ 4 dim(E1 ) + 3. Consider the set C = N (η/2)\(E ′ + N (η/4)). If C ̸= ∅, pick a point x0 ∈ E\(E ′ + N (η/4)) to have (2)

∥x0 ∥ < η/2.

Otherwise, there exists a point x0 ∈ E\E ′ such that (2’)

∥sx0 ∥ < η/2 for every s ≥ 0.

In either case, let E0 = span(E ′ , {x0 }). By a theorem of Michael, see C.Bessaga, A.Pelczy´ nski [1975, p.85], there exists a continuous right inverse ¯ 0→E ¯ for the quotient map κ : E ¯ → E/E ¯ 0 ; by the local convexity φ : E/E of E we may additionally require that (3)

¯ ∥φ ◦ κ(x)∥ ≤ 2∥x∥ for all x ∈ E.

Writing r(x) = x − φ ◦ κ(x), by (1) and (3), we have (4)

r(A) ⊂ E ′ + N (η/8).

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The formula t ∈ [0, 1], b + tx0 , αt (b) = (2 − t) · (b + x0 ) + (t − 1) · (f (b) + x0 ), t ∈ [1, 2], f (b) + (3 − b)x0 , t ∈ [2, 3], deﬁnes a homotopy joining id|B with f in E0 . By (2), we have (5)

∥αt (b) − b∥ < η/2 + ∥f (b) − b∥ + η/2 < ε + ∥f (b) − b∥

for every b ∈ B. Moreover, by (4), (2) or (2’) we get (6)

α(b × [0, 3]) ∩ r(A) = ∅.

(In the case where C = ∅, in the formula describing α we put instead of x0 , the point sx0 in order to have (sx0 + E ′ ) ∩ r(A) = ∅). By 5.4.9, there exists ¯ ∈ H(E0 ) with h|B ¯ = f satisfying, by (5), a homeomorphism h ¯ − id∥ < ε + ∥f − idB ∥ ∥h ¯ and such that, by (6), h|r(A) = id|r(A). Finally, we put h(x) = φ ◦ κ(x) + ¯ ¯ ¯ E). h ◦ r(x) for x ∈ E. Clearly, h ∈ H(E, Proof of 5.4.8. Assume that δ < dist(A, B) and consider a ﬁnite-dimensional linear space E0 ⊂ E with (7)

A ∪ B ⊂ E0 + N (η/4).

¯ 0 →E ¯ be a map of the proof of 5.4.7 (i.e. φ satisﬁes the Let φ : E/E ¯ condition (3)). Write h0 (x) = (κ(x), r(x)) for the homeomorphism of E ¯ onto E/E0 × E0 with r(x) = x − φ ◦ κ(x). Let ∥ · ∥ be the quotient F -norm on E/E0 = Y . By (7), we have r(A) ∩ r(B) = ∅. Let ω : E0 → [0, δ/4] be a map with ¯ h0 (B) ⊂ {(y, e) ∈ Y × E0 : ∥y∥ < ω(e)} ⊂ h0 (E\A). Let {αt : [0, ∞] → (0, 1]}, 0 ≤ t < 1, be a homotopy of [0, ∞] with α0 ≡ 1, αt is monotone and αt |[1, ∞] = 1 for each t such that for each s < 1, limt→1 αt (s) = 0. Deﬁne an isotopy {gt }, 0 ≤ t < 1, of Y × E0 by the formula { (αt (∥y∥/ω(e)) · y, e), if y ̸= 0, gt (y, e) = (0, e), if y = 0. Then the desired map u may be deﬁned by u(x, t) = h−1 0 gt h0 (x) for 0 ≤ t < 1, and u(b, 1) = r(b) for b ∈ B.

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Proof of 5.4.6. (1) Assume span(B ∪ f (B)) = E1 ⊂ E is ﬁnite-dimensional. One can easily approximate f by maps with ranges disjoint from A (since ¯ Thus, we will require that f (B)∩A = ∅. A, being compact, is a Z-set in E). Since B is ﬁnite-dimensional, there exists an embedding w : B → E with ∥w(b)∥ < δ and such that E2 = span(w(B)) is ﬁnite-dimensional with E2 ⊂ E and E2 ∩ E1 = {0}. Writing v = f + w, we see that v is an embedding satisfying (8)

∥v − id|B∥ < ∥f − id∥ + δ and ∥v − f ∥ < δ.

If δ is suﬃciently small, we also get (9)

v(B) ∩ A = ∅.

Assuming δ < ε and applying Lemma 5.4.7, we extend the embedding v to a required homeomorphism h. ¯→E ¯ be a map with f¯|B = f (we use the (2) The general case. Let f¯ : E ¯ Pick δ with 0 < δ < ε/8 and such that AR-property of E). (10)

b ∈ B, ∥x − b∥ < δ implies ∥f¯(x) − f (b)∥ < ε/4.

With this δ, consider a homotopy ut of Lemma 5.4.8 and write B ′ = u1 (B). Let us approximate f¯|B ′ by f ′ : B ′ → E such that (11)

∥f¯|B ′ − f ′ ∥ < ε/4 and dim span(f ′ (B ′ )) < ∞.

Now observing that the triple (B ′ , A, f ′ ) satisﬁes the assumptions of the ¯ E) satisfying ∥h′ |B ′ − previous case, we conclude that there exists h′ ∈ H(E, ′ ′ ′ ′ ′ f ∥ < ε/4, ∥h − id∥ < ∥f − id|B ∥ + ε/4 and h |A = id|A. We claim that ht = h′ ◦ ut may serve as a required homeomorphism, provided that t ̸= 1 is suﬃciently close to 1. First observe that for b′ = u1 (b) ∈ B ′ , we can estimate, using (11), (v), and (1), ∥f ′ (b′ ) − b′ ∥ ≤ ∥f ′ (b′ ) − f¯(b′ )∥ + ∥f¯ ◦ u1 (b) − f (b)∥ + ∥f (b) − b∥+ + ∥b − u1 (b)∥ ≤ ε/4 + ε/4 + ∥f − id|B∥ + ε/8. Consequently, by (v), for all t < 1, we have ∥ht − id∥ ≤ ∥h′ ◦ ut − ut ∥ + ∥ut − id∥ ≤ ∥h′ − id∥ + ε/8 ≤ ≤ 5ε/8 + ∥f − id|B∥ + ε/4 + ε/8 ≤ ∥f − id|B∥ + ε. Now we take t0 ∈ (0, 1) such that for every t ≥ t0 we have (12)

∥h′ ◦ ut (b) − h′ ◦ u1 (b)∥ < ε/4 for all b ∈ B.

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171

Using (12)–(10), for t ∈ [t0 , 1), we have ∥ht (b) − f (b)∥ = ∥h′ ut (b) − f (b)∥ ≤ ≤ |h′ ut (b) − h′ u1 (b)∥ + ∥h′ u1 (b) − f ′ u1 (b)|| + ∥f ′ u1 (b) − f¯u1 (b)∥+ + ∥f¯u1 (b) − f (b)∥ < ε/4 + ε/4 + ε/4 + ε/4. Finally, since both h′ and ut are the identity on A, the composition ht = h′ ◦ ut has this property also. ¯ and a map f : A → E ¯ Proof of 5.4.1. Fix a positive ε, a compactum A ⊂ E, such that f |B is a Z-embedding with (f |B)−1 (E) = B ∩ E. Replacing A be A + e for suitable e ∈ E, if necessary, we may assume that A∩f∪ (A) = ∅. Consider a tower {Ai }∞ i=0 consisting of compacta, with A0 = ∅ ∞ and i=0 Ai = A\B. We shall inductively construct a sequence of homeo¯ morpﬁsms {hn }∞ n=0 ⊂ H(E, E) such that, writing gn = hn ◦ hn−1 ◦ · · · ◦ h0 , the following conditions are satisﬁed: (13)

∥gn − gn−1 ∥ < 2−n+1 ε

(14)

gn |An−1 = gn−1 |An−1 for n ∈ N;

(15)

∥gn |B − f |B∥ < 2−n−1 ε

and (16)

gn |f (A) = id for n = 0, 1, 2, . . . .

Let h0 be a homeomorphism obtained by Lemma 5.4.6 applied to the quadruple (A, f, f (A), ε/2). Assume h0 , . . . , hn−1 (n ≥ 1) are already constructed. To get hn , apply Lemma 5.4.6 to the quadruple −1 (gn−1 (B), f ◦ gn−1 |gn−1 (B), gn−1 (An ) ∪ f (A), 2−n−1 ε).

Then (13) is a consequence of the calculation ∥gn − gn−1 ∥ = ∥hn − id∥ < −1 < ∥f ◦ gn−1 |gn−1 (B) − id|gn−1 (B)∥ + 2−n−1 ε ≤

≤ ∥f |B − gn−1 |B∥ + 2−n−1 ε ≤ 2−n ε + 2−n−1 ε < 2−n+1 ε. The conditions (14)–(16) also follow. Now put g(a) = lim gn (a) for a ∈ A. By (13)–(15), g is a continuous ¯ such that g|B = f |B and ∥g − f ∥ < 3ε. Finally, by (14) map from A into E ¯ is a and (16), g is a 1-1 map and g −1 (E) = E. Since each compactum in E Z-set, g is a Z-embedding.

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Exercises and Problems to §5.4. 1. (Open Problem) Is every locally convex (Zσ -) space strongly Z(X)-universal? 2. (Open Problem) Is the set {h ∈ H(l2 ) | x − h(x) ∈ lf2 for all x ∈ l2 } dense in H(l2 ) equipped with the limitation topology? Remark that this set is dense if H(l2 ) is endowed with the compact-open topology, see T.Dobrowolski [1989]. 3. Show that every homeomorphism between closed locally compact subsets of an inﬁnitedimensional locally convex space E can be extended to a homeomorphism of whole E. Hint: see T.Dobrowolski [1989]. 4. Generalize the results of §§5.3, 5.4 to pairs and Γ-systems of convex sets.

§5.5. Some counterexamples In this section we will construct: 1) a pre-Hilbert space that is not strongly universal, 2) a normed space that is absorbing, but not homeomorphic to any convex set in l2 , and 3) a strongly universal pre-Hilbert space that is a Borel space of the ﬁrst Baire category, but not a Zσ -space. A. An example of a pre-Hilbert space that is not strongly universal. This example is due to J. van Mill [1987] and has a lot of other unexpected properties. Construction. Let X be a Banach space and let K(X) denote the collection of all homeomorphisms h : K1 → K2 between disjoint Cantor sets in X such that K1 ∪ K2 is a linearly independent set. 5.5.1. Theorem. Every inﬁnite-dimensional Banach space X has a linear subspace Y with the following property: (∗)

∀h ∈ K(X), ∃x ∈ dom h such that x ∈ Y but h(x) ∈ / Y.

Proof. It is easy to see that the collection K(X) has size at most c. Let < be a well-ordering on K = K(X) such that for each h ∈ K, the section {g ∈ K | g < h} has size less than c. We show by transﬁnite induction that for all h ∈ K, there exist linear subspaces Yh and Zh in E satisfying the following conditions. • Yh ∩ Zh = {0}; for g < h, Yg ⊂ Yh and Zg ⊂ Zh ; • all Yh and Zh have algebraic dimension ≤ |{g ∈ K | g < h}|, and • there exists x ∈ dom h such that x ∈ Yh and h(x) ∈ Zh . ∪ Then Y = {Yh | h ∈ K} is a linear subspace with the required property (∗). For the ﬁrst element f ∈ K, we may take Yf = span{x} and Zf = span{f (x)} for any x ∈ dom f . For h ∈ K,∪suppose the spaces Yg and ∪ Zg h h have been constructed for g < h. Let Y = {Yg | g < h} and Z = {Zg |

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173

g < h}. Then Y h ∩ Z h = {0}, and span(Y h ∪ Z h ) has algebraic dimension less than c. Consider H = {x ∈ dom h | span({x} ∪ Y h ) ∩ span({h(x)} ∪ Z h ) ̸= {0}}. We claim that H has size less than c. For each x ∈ H, there exist scalars λ(x) and µ(x) such that w(x) = λ(x) · x + µ(x) · h(x) ∈ span(Y h ∪ Z h )\{0}. Thus either λ(x) ̸= 0 or µ(x) ̸= 0. To establish the claim on H, we show that the correspondence of x 7→ w(x) is injective and has linearly independent range {w(x) | x ∈ H}. Consider any ﬁnite, faithfully indexed subset {x1 , . . . , xn } of H. Let I = {i | λ(xi ) = 0} and J = {i | λ(xi ) ̸= 0}. Then span{xi , h(xi )} = span{xi , w(xi )} for i ∈ I, and span{xi , h(xi )} = span{h(xi ), w(xi )} for i ∈ J. Hence the set {xi | i ∈ I} ∪ {h(xi ) | i ∈ J} ∪ {w(xi ) | 1 ≤ i ≤ n} has the same span as the set {xi | 1 ≤ i ≤ n} ∪ {h(xi ) | 1 ≤ i ≤ n}. The stipulated properties of h ∈ K imply that the latter set has size 2n and is linearly independent. It follows that {w(xi ) | 1 ≤ i ≤ n} has size n and is linearly independent. Since {xi | 1 ≤ i ≤ n} ⊂ H was arbitrary, this completes the argument for the claim on H. Since H has size less than the size of the Cantor set dom h, there exists x ∈ dom h\H, i.e. span({x} ∪ Y h ) ∩ span({h(x)} ∪ Z h ) = {0}. The induction step is completed by setting Yh = span({x} ∪ Y h ) and Zh = span({h(x)} ∪ Z h ). 5.5.2. Theorem. The space Y constructed in 5.5.1 has the following properties: (1) Y is dense in X; (2) Y is a Baire space; (3) for every map f : Y → Y there is a countable set Z ⊂ Y such that f (y) ∈ span({y} ∪ Z) for all y ∈ Y ; (4) every convex compact subset of Y is ﬁnite-dimensional; (5) if U ⊂ Y is open then for every injective map f : U → Y there is a dense open V ⊂ U such that f |V : V → Y is an open embedding; (6) Y admits no Z-embedding Y → Y , and thus Y is not strongly F0 (Y )-universal; (7) Y is not homeomorphic to Y × Z for every nondegenerate space Z.

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Proof. We need to introduce at ﬁrst some notation. If a function f : Y → Y on a linear space satisﬁes the condition (3) of 5.5.2, we say that f has countable type. Let X be a linear space, and n a positive integer. A ﬁnite subset ∑k {x1 , . . . , xk } of E is said to be n-linearly essential if i=1 λi xi ̸= 0 for all λi such that 1/n ≤ |λi | ≤ n for each i. The following lemma is trivial; nonetheless it will be quite useful for establishing linear independence of a set which is constructed by “approximation” to a sequence of linearly independent sets. 5.5.3. Lemma. (1) A subset K in a linear space is linearly independent if and only if each ﬁnite subset of K is n-linearly essential for each n. (2) If {x1 , . . . , xk } is an n-linearly essential subset of a metric space, there exists ε > 0 such that, if d(xi , yi ) < ε for each i, then the set {y1 , . . . , yk } is also n-linearly essential. Let X be a linear space, and g : A → E a function deﬁned on a subset of X. A subset P of A is said to be g-independent if the following conditions are satisﬁed: (i) g|P is injective; (ii) P ∩ g(P ) = ∅; and (iii) P ∪ g(P ) is linearly independent. 5.5.4. Lemma. Let X be a metric linear space, and g : A → X a map deﬁned on a separable, topologically complete subset. If A contains an uncountable g-independent set, then A contains a g-independent Cantor set. Proof. Let d be a metric on X, and choose a complete metric ρ on A. For each x ∈ A and ε > 0, let B(x, ε) = {a ∈ A | ρ(a, x) ≤ ε}. Since each separable metric space is the union of a countable set and a perfect set (each point is limit), the hypothesis implies that A contains a perfect gindependent set P . Using ﬁnite disjoint unions of balls about points of P , we may construct a Cantor set K in the complete space A by the standard procedure; a little extra care will ensure that K is g-independent. It suﬃces to describe the ﬁrst two steps in the inductive construction. Pick any p1 ∈ P . Since g(p1 ) ̸= p1 , there exists 0 < ε1 < 1 such that B(p1 , ε1 ) ∩ g(B(p1 , ε1 )) = ∅. Let B1 = B(p1 , ε1 ). Since the set {p1 , g(p1 )} is linearly independent, we may assume by Lemma 5.5.3(2) that ε1 is sufﬁciently small so that, for any F ⊂ B1 ∪ g(B1 ) such that F contains at most a single point from each of B1 and g(B1 ), F is 1-linearly essential. Let K1 = B1 . Since P is perfect, there exist distinct points p1,0 and p1,1 in P ∩ B1 . Choose 0 < ε2 < 21 such that, for B1,0 = B(P1,0 , ε2 ) and B1,1 = B(p1,1 , ε2 ), we have B1,0 ∪ B1,1 ⊂ B1 , B1,0 ∩ B1,1 = ∅, and g(B1,0 ) ∩ g(B1,1 ) = ∅. Since the set {p1,0 , p1,1 , g(p1,0 ), g(p1,1 )} is linearly independent and has size 4, we may also assume that ε2 is small enough so that, for any F ⊂ B1,0 ∪ B1,1 ∪

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g(B1,0 ) ∪ g(B1,1 ) such that F contains at most a single point from each of B1,0 , B1,1 , g(B1,0 ), and g(B1,1 ), F is 2-linear essential. Se K2 = B1,0 ∪B1,1 . Continuing with this procedure in the standard∩manner, we obtain a ∞ nested sequence (Kn ) of closed sets in A. Let K = n=1 Kn . The requirements of the type B1,0 ∪ B1,1 ⊂ B1 and B1,0 ∩ B1,1 = ∅, together with the requirements that εn → 0 and the fact that ρ is a complete metric, show that K is a Cantor set. The requirements of the type g(B1,0 ) ∩ g(B1,1 ) = ∅ show that g|K is injective. Since g(K) ⊂ g(B1 ) and K ⊂ B1 , K ∩ g(K) = ∅. And ﬁnally, the n-linearly essential requirements at the n-th stage of construction ensures that each ﬁnite subset of K ∪ g(K) is n-linearly essential for each n, hence K ∪g(K) is linearly independent by Lemma 5.5.3(1). Thus K is g-independent. 5.5.5. Remark. If we assume only that A contains an uncountable linearly independent set, the above construction shows that A contains a linearly independent Cantor set. 5.5.6. Lemma. A function f : Y → Y on a linear space Y has countable type if and only if it satisﬁes the following conditions: (i) every f -independent set is countable; and (ii) for every countable set P ⊂ Y , there exists a countable set Pˆ ⊂ Y such that f (span P ) ⊂ span Pˆ . Proof. Suppose ﬁrst that f has countable type; let Z be a countable subset of Y such that f (y) ∈ span({y} ∪ Z) for each y. Let T = {y ∈ Y | f (y) ∈ / span{y}}. Then for each y ∈ T , span Z ∩ span{y, f (y)} ̸= {0}. Consider any f -independent set A. We have A ⊂ T , and for each a ∈ A we may choose sa ∈ (span Z ∩ span{a, f (a)})\{0}. Since A is f -independent, the set {sa | a ∈ A} is linearly independent subset of span Z, and is therefore countable. It follows that A is countable, and conditions (i) is satisﬁed. Condition (ii) is clear, since f (span P ) ⊂ span(P ∪ Z) for every P ⊂ Y . Conversely, assume conditions (i) and (ii). It is easily seen that in the collection of all f -independent subsets, partially ordered by inclusion, every chain has an upper bound. Thus there exists a maximal f -independent set P1 , which by hypothesis is countable. If P1 = ∅, then f (y) ∈ span{y} for each y ̸= 0, and f obviously has countable type. Otherwise, construct a tower∪(Pi ) of countable sets by taking Pn+1 = Pn ∪ Pˆn for n ≥ 1. Take ∞ Z = n=1 Pn . Then Z is countable, P1 ⊂ Z and f (span Z) ⊂ span Z. We claim that f (y) ∈ span({y} ∪ Z) for each y. This is clear for y ∈ span Z so consider y ∈ / span Z. The set P1 ∪ {y} cannot be f -independent, and one of the following occurs: (1) f (y) = f (p) for some p ∈ P1 . Then f (y) ∈ span Z.

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(2) y = f (p) for some p ∈ P1 . Then y ∈ span Z. (3) {y, f (y)} ∪ P1 ∪ f (P1 ) is not linearly independent. In this case, either f (y) ∈ span({y} ∪ P1 ∪ f (P1 )) ⊂ span({y} ∪ Z), or y ∈ span(P1 ∪ f (P1 )) ⊂ span Z. Thus f has countable type.

Now we are able to prove the conditions (1)—(4) for the space Y from 5.5.2. We show ﬁrst that Y intersects every dense Gδ -subset of X. Every such set A must contain an uncountable linearly independent set, since otherwise span A is an ℵ0 -dimensional linear subspace, and X = span(A) ∪ (X\A) would be ﬁrst category. Then A contains a linearly independent Cantor set K (see 5.5.5). Let h : K1 → K2 be any homeomorphism between disjoint Cantor subsets of K. Then h ∈ K(X), so Y ∩ A ⊃ Y ∩ K1 ̸= ∅. In particular, Y¯ must have nonempty interior in X, which implies that ¯ Y = X. Further, the fact that Y intersect every dense Gδ -subset of X means that Y is second category, which implies Y is a Baire space. We now show that every map f : Y → Y has countable type. Since X is complete, f extends to a map g : A → X for some Gδ -subset A of X. Suppose that A contains an uncountable g-independent set. By Lemma 5.5.4, A contains a g-independent Cantor set K. Then g|K is a member of the collection K(X), and by hypothesis there exists x ∈ K ∩ Y such that g(x) ∈ / Y . But this contradicts the fact that g(x) = f (x) ∈ Y . Thus every g-independent subset of A is countable, and in particular, every f independent set is countable. Therefore, condition (i) of Lemma 5.5.6 is satisﬁed. It remains to verify condition (ii). For any countable set P ⊂ Y , span P is σ-compact, and f (span P ) is σ-compact. If some compactum in Y contains an uncountable linearly independent set, then Y contains a linearly independent Cantor set, but this contradicts property (∗). Thus each compactum in Y lies in an ℵ0 -dimensional linear subspace, and f (span P ) ⊂ span Pˆ for some countable set Pˆ . By Lemma 5.5.6 the condition (3) follows. Since every compactum K in Y is strongly countable-dimensional, K can not be a Keller cube. Thus (4) holds. The properties (6), (7) trivially follows from (5). To prove the condition (5), notice that since Y admits an open embedding into any open subset U ⊂ Y , it suﬃces to ﬁnd for a given injective map f : Y → Y an open set V ⊂ Y such that f |V is an open embedding. Let Z ⊂ Y be a countable set such that f (y) ∈ span({y}∪Z) for each y. There is a tower of compacta (An ) ∪∞ such that span An is ﬁnite-dimensional for each n, and n=1 An = span Z. For each n, set Yn = {y ∈ Y | for some λ ∈ [−n, n], f (y) − λ · y ∈ An }.

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∪∞ By compactness of [−n, n] and An , each Yn is closed in Y . Since n=1 Yn = Y , and Y is a Baire space, some Yn has nonempty interior. Since span An is ﬁnite-dimensional, there exists a nonempty open set W ⊂ Yn \ span An . Then for each w ∈ W , there is a unique λ(w) ∈ [−n, n] such that f (w) − λ(w) · w ∈ An (otherwise, W ∩ span An ̸= ∅). Furthermore, the compactness of [−n, n] and An , and the continuity of f , show that the assignment w 7→ λ(w) is continuous. Let λ : W → [−n, n] denote this map, and let α : W → An denote the map deﬁned by α(w) = f (w) − λ(w) · w. If λ(W ) = {0}, we have f (W ) ⊂ An . But this is impossible, since f is injective, the open set W contains cells of all dimensions, and An is ﬁnite-dimensional. Thus, there exists a nonempty open V ⊂ W with either λ(V ) ⊂ [−n, 0) or λ(V ) ⊂ (0, n]. In either case, we apply the following lemma to conclude that f |V is an open embedding. 5.5.7. Lemma. Let Y be a normed linear space, F a ﬁnite-dimensional linear subspace, and V ⊂ Y \F an open subset. Let λ : V → (0, ∞) and α : V → F be maps such that the map f : V → Y deﬁned by f (v) = λ(v) · v + α(v) is injective. Then f is an open embedding. Proof. Since the restriction f |W to any open W ⊂ V is a map with the same properties, it suﬃces to show that f (V ) is open. Consider an arbitrary point p ∈ V . We may assume that λ(p) = 1 and α(p) = 0, thus f (p) = p. Let E = span({p} ∪ F ). Note that f (E ∩ V ) ⊂ E. Choose δ > 0 and a compact neighborhood B of 0 in F such that D = [1 − δ, 1 + δ] · p + B ⊂ V . Then D is a compact neighborhood of p in E. Since f is injective and E is ﬁnite-dimensional, f |D : D → E is an embedding and f (D) is a neighborhood of f (p) = p in E (Brouwer domain invariance principle). Thus p is a stable point of f (D), and any map f˜ : D → E which is suﬃciently close to f |D must cover p. For each v ∈ V , let Ev = span({v} ∪ F ), and deﬁne the homeomorphism hv : E → Ev by hv (t · p + s) = t · v + s for t ∈ R and s ∈ F . Let Dv = hv (D) = [1 − δ, 1 + δ] · v + B. Then for all v, near p, Dv ⊂ V and f (Dv ) ⊂ Ev . Deﬁne a map f˜v : D → E ˜ by f˜v = h−1 v ◦ f ◦ hv |D. Then fv → f |D as v → p. Hence for all v ˜ suﬃciently close to p, p ∈ fv (D), which means that v ∈ f (Dv ). Thus f (V ) is a neighborhood of p = f (p). B. On topological embeddings of locally convex spaces. In this subsection we are concerned with the following question: when a given linear metric space is homeomorphic to a convex subset of a “nice” Banach space (such as l2 ). Clearly, Cauty’s example of a σ-compact linear

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metric space, which is not an AR, supplies us with an example of a σcompact linear metric space, homeomorphic to no convex set in a locally convex space (remark also that the Cauty space can be linearly embedded into a linear metric AR-space, see Ex.14 to §4.2). On the other hand, by Theorem 5.4.4, every σ-precompact locally convex space is homeomorphic to a pre-Hilbert space. Examples, constructed in this section, show that this result cannot be extended beyond the class of σ-precompact spaces. Namely, we prove the following two theorems. 5.5.8. Theorem. The Banach space l1 contains a dense linear subspace Y such that (1) (2) (3) (4)

Y is an absorbing space; Y ∼ = Y × σ; Y ̸∼ =Y ×Y; no convex subspace of a reﬂexive Banach space is homeomorphic to Y.

5.5.9. Theorem. The Fr´echet space Rω contains a dense linear subspace Y such that (1) (2) (3) (4)

Y is an absorbing space; Y ∼ = Y × σ; Y ∼ ̸ Y ×Y; = no convex subspace of a Banach space is homeomorphic to Y .

For the proof of these theorem we need to introduce some deﬁnition. Let X, Y be linear metric spaces and let f : C → Y be an injective map deﬁned on a convex subset C ⊂ X. We say that f preserves segments, if for every a, b ∈ C f ([a, b]) = [f (a), f (b)]. It is easy to observe that in this case the image of every convex subset of C is convex (in particular, f (C) is convex), and f −1 : f (C) → C also preserves segments. Also for every A ⊂ C we have f (C ∩ aﬀ A) = f (C) ∩ aﬀ f (A). In the sequel we will need the following fact. 5.5.10. Lemma. Let E be a locally convex space. Let M , N , and S be subsets of E, which are n-dimensional manifolds. If M and N are connected and the algebraic sum 12 M + 12 N is contained in S then M and N are “ﬂat”, i.e. there exist n-dimensional aﬃne subspaces P and P ′ in E such that M ⊂ P and N ⊂ P ′ . Proof. It is easy to observe that by connectedness it is enough to show that all vectors x ∈ M and y ∈ N have neighborhoods U ⊂ M and V ⊂ N which are “ﬂat”. We may assume that x = y = 0. Then 12 M ⊂ S and 12 N ⊂ S and M, N ⊂ 2S. Since M , N and 2S are n-dimensional manifolds, it follows that M and N must coincide on some neighborhood of 0. Therefore, we may

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restrict ourselves to the case when M = N . Then 21 M + 12 M ⊂ S which, in particular, mean that M ⊂ S. Let V be a closed convex neighborhood of 0 in E such that U = V ∩ S is compact and U ⊂ M . Then 12 U + 12 U ⊂ S and 1 1 1 1 2 U + 2 U ⊂ V by the convexity of V . Hence 2 U + 2 U ⊂ V ∩ S = U , which together with the compactness of U implies that U is convex. Then U , being a convex n-dimensional set, must be contained in some n-dimensional linear subspace P of E. Before we state our main lemma we need to deﬁne a certain auxiliary and rather technical property which will be used in the proofs of our theorems. 5.5.11. Definition. Let X and Y be linear metric spaces. We say that the pair (X, Y ) has the property (⋆) if X contains a linear subspace Z such that the closure Z¯ has ﬁnite codimension in X and for some nonempty open convex subset U ⊂ Z there exists a homeomorphic embedding h : U → Y preserving segments. Notice that if a pair (E, F ) does not have the property (⋆) then E is inﬁnite-dimensional and for every dense linear subspace X ⊂ E the pair (X, F ) does not have (⋆) neither. 5.5.12. Lemma. Let E and F be inﬁnite-dimensional Fr´echet spaces, E0 a closed linear subspace of E such that the pair (E0 , F ) does not have the property (⋆), and let h : A → B be a homeomorphism between Gδ -subsets A and B of E and F , respectively. If A contains a linear subspace X of E such that E0 ∩ X is dense in E0 and h(X) is convex, then A ∩ E0 contains a topological copy C of the Cantor set with the following properties: a) for every x, y ∈ C we have h(x)+h(y) ∈ B; 2 b) for every sequence x1 , y1 , x2 , y2 , . . . , xn , yn of distinct elements of C the 2) 1) ), x2 , y2 , h−1 ( h(x2 )+h(y ), . . . , xn , yn , vectors x1 , y1 , h−1 ( h(x1 )+h(y 2 2 n) h−1 ( h(xn )+h(y ) are linearly independent. 2 For constructing the set C we will use the following fact from Ramsey theory of Polish spaces, see A.Kechris [1995,19.1]. 5.5.13. Theorem (Mycielski, Kuratowski). Let A be a dense-in-itself Polish space and for every n ∈ N let Rn ⊂ An be a dense Gδ -subset. Then A contains a Cantor cube C such that {(ci ) ∈ C n | ci ̸= cj for i ̸= j} ⊂ Rn for every n ∈ N. Proof of 5.5.12. Obviously, A ∩ E0 is a dense-in-itself Polish space. Let . We put g : A × A → F be a map deﬁned by g(x, y) = h(x)+h(y) 2 R0 = {(x, y) ∈ (A ∩ E0 )2 | x ̸= y and g(x, y) ∈ B}.

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One can easy verify that R0 is a Gδ -set containing (X ∩ E0 )2 ; therefore, it is dense in (A ∩ E0 )2 . For every n ≥ 1 let Rn = {((x1 , y1 ), . . . , (xn , yn )) ∈ R0n ⊂ (A ∩ E0 )2n | the vectors x1 , y1 , h−1 g(x1 , y1 ), . . . , xn , yn , h−1 g(xn , yn ) are linearly independent. It is easily seen that each Rn is open in R0n . We will show, by induction on n, that Rn is also dense in R0n . Then applying Theorem 5.5.13, we ﬁnd a Cantor set C in A ∩ E0 with the required properties. Suppose either n = 1 or Rn−1 is dense in R0n−1 . Fix a sequence U1 , V1 , . . . , Un , Vn of non-empty open sets of E0 . We shall ﬁnd (x1 , y1 , . . . , xn , yn ) ∈ Rn such that xi ∈ Ui and yi ∈ Vi for i = 1, . . . , n. First, we take (x1 , y1 , . . . , xn−1 , yn−1 ) ∈ Rn−1 with xi ∈ Ui and yi ∈ Vi for i = 1, . . . , n − 1. Let G = span{x1 , y1 , h−1 g(x1 , y1 ), . . . , xn−1 , yn−1 , h−1 g(xn−1 , yn−1 )} (G = {0} if n = 1). For x, y, z ∈ X we deﬁne G(x, y, z) = span({x, y, z} ∪ G) ⊂ X and G(x, y) = G(x, y, x). Reﬁning Un and Vn , if necessary, we may assume that they are convex and for every x ∈ Un and y ∈ Vn we have (span{x, y})∩ G = {0} and x, y are linearly independent. To end the inductive step it is enough to show that there exist x ∈ Un ∩X and y ∈ Vn ∩ X with the property that g(x, y) ∈ / h(G(x, y)) (then we can take xn = x and yn = y). Suppose the contrary (we will get the contradiction by proving that (E0 , F ) has property (⋆)). Then for every x ∈ Un ∩ X, y ∈ Vn ∩ X, z ∈ X, x′ ∈ Un ∩ G(x, y, z) and y ′ ∈ Vn ∩ G(x, y, z) we have g(x′ , y ′ ) ∈ h(G(x′ , y ′ )) ⊂ h(G(x, y, z)). Therefore the sets M = h(Un ∩ G(x, y, z)), N = h(Vn ∩ G(x, y, z)) and S = h(G(x, y, z)) satisfy the assumptions of Lemma 5.5.10. It follows that h(Un ∩ G(x, y, z)) is contained in some aﬃne subspace P (x, y, z) ⊂ F of dimension equal to dim G(x, y, z). In particular, the above statement holds for G(x, y) = G(x, y, x), and we put P (x, y) = P (x, y, x). Fix x ¯ ∈ Un ∩ X and y¯ ∈ Vn ∩ X. For x, x′ ∈ Un let ix,x′ : G(x, y¯) → G(x′ , y¯) be the linear homeomorphism which is the identity on span({¯ y } ∪ G) and ix,x′ (x) = x′ . Without loss of generality we may assume that h(¯ x) = 0, so h(Un ∩ G(¯ x, y¯)) is contained in the linear subspace F1 = P (¯ x, y¯) ⊂ F of the same dimension as G(¯ x, y¯). We take a closed linear subspace F2 ⊂ F such that F1 is complemented in F by F2 , i.e. F = F1 ⊕ F2 . Let p : F → F1 be the continuous projection with p−1 (0) = F2 .

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For every x ∈ Un we put O(x) = h(Un ∩ G(x, y¯)). Observe that O(x) is an open subset of P (x, y¯). Let U = {x ∈ Un | O(x) intersects F2 at exactly one point}. Obviously, x ¯ ∈ U ; we will show that U is open in X and the map f : U → F deﬁned by {f (x)} = O(x) ∩ F2 is continuous. Fix x ∈ U , then p|O(x) is a homeomorphic embedding onto some neighborhood of 0 in F1 . Take x′ ∈ Un ∩ G(x, y¯) such that h(x′ ) = f (x), i.e. p◦h(x′ ) = 0. Let W be an arbitrary neighborhood of f (x) in F . We can ﬁnd a compact neighborhood K of x′ in Un ∩ G(x, y¯) such that h(K) ⊂ W ; then p ◦ h(K) is a compact neighborhood of 0 in F1 . Since dim G(x, y¯) = dim F1 we have 0 ∈ Int q(K) for every map q : K → F1 suﬃciently close to p ◦ h|K. Now ﬁnd a neighborhood U ′ of x in X such that U ′ ⊂ Un , ix,u (K) ⊂ U , and h ◦ ix,u (K) ⊂ W , for every u ∈ U ′ . We may also assume that p ◦ h ◦ ix,u |K is suﬃciently close to p ◦ h|K, hence p ◦ h ◦ ix,u (K) is a neighborhood of 0. It follows that u ∈ U and f (u) ∈ h ◦ ix,u (K) ⊂ W , for every u ∈ U ′ , which gives the required properties of U and f . Let Y be a closed linear subspace of X ∩ E0 such that X ∩ E0 = G(¯ x, y¯) ⊕ Y . Since X ∩ E0 is dense in E0 , we get E0 = G(¯ x, y¯) ⊕ Y¯ . We may assume that G(x, y¯) ∩ Y = {0}, for every x ∈ Un . Then G(x, y¯) intersects x ¯ + Y at exactly one point x ¯ + r(x), where r(x) ∈ Y . Similarly as for the map f one may check that the map r : Un → Y is continuous. Let V be a convex open neighborhood of 0 in Y such that x ¯ + V ⊂ U. We deﬁne the map e : V → F by e(z) = f (¯ x + z), for z ∈ V . From our assumptions on Un and Vn it follows that Un ∩ span({¯ y } ∪ G) = ∅. Therefore, for every z, z ′ ∈ V , z ̸= z ′ , we have (Un ∩ G(¯ x + z, y¯)) ∩ (Un ∩ G(¯ x + z ′ , y¯)) = ∅ and O(¯ x + z) ∩ O(¯ x + z ′ ) = ∅. This shows that e in −1 one-to-one. It is easy to verify that e (y) = r ◦h−1 (y), for y ∈ e(V ), hence, e is a homeomorphic embedding. We will show that e preserves segments. Let z, z ′ ∈ V . Then we have h(Un ∩ G(¯ x + z, x ¯ + z ′ , y¯)) ⊂ P (¯ x + z, x ¯ + z ′ , y¯) ′ and the intersection P (¯ x + z, z¯ + z , y¯) ∩ F2 is at most one dimensional. The last follows from the fact that P (¯ x + z, y¯) is of codimension ≤ 1 in P (¯ x + z, x ¯ + z ′ , y¯) and that the open set O(¯ x + z) ⊂ P (¯ x + z, y¯) intersects F2 at exactly one point. Since e([z, z ′ ]) is contained in this intersection and e is a homeomorphic embedding, we conclude that e([z, z ′ ]) = [e(z), e(z ′ )]. This prove the property (⋆) for (E0 , F ), a contradiction. The proofs of 5.5.8, 5.5.9 are based on the following general fact: 5.5.14. Theorem. Every Fr´echet space E contains a dense linear subspace X such that for every Fr´echet space F such that (E, F ) does not have the property (⋆), no convex subspace of F is homeomorphic to X × lf2 . Proof. Below we identify E × l2 with E ⊕ l2 . Recall that C(I) denotes the Banach space of all real-valued continuous maps on [0, 1].

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Without loss of generality, we may assume that there is an inﬁnitedimensional Fr´echet space F such that (E, F ) does not have the property (⋆) (otherwise our theorem is a triviality). In this case E and F are inﬁnite-dimensional, and hence, by Anderson-Kadeˇc Theorem 1.1.27, they are homeomorphic. Let H be the family of all homeomorphisms h : A → B such that a) A is a Gδ -subset of E ⊕ l2 ; b) B is a Gδ -subset of C(I)ω such that (E, span B) does not have the property (⋆); c) there is a dense linear subspace Y ⊂ E such that Y ⊕lf2 ⊂ A and h(Y ⊕lf2 ) is convex. Observe that the family H has the size of the continuum c. Let < be a well-ordering on H such that for each h ∈ H the section {g ∈ H | g < h} ˜ = has size less than c. Let D be a countable dense subset of E and let D D ∪ {en | n ∈ N}, where (en ) stands for the standard orthonormal basis of l2 (and thus lf2 = span{en }). By transﬁnite induction we will chose vectors xα , yα ∈ E, zα ∈ E ⊕ l2 , for α ∈ H, in such a way that the following conditions will be satisﬁed for every h ∈ H: ˜ for g ≤ h, (i) zg ∈ / span({xf , yf | f ≤ h} ∪ D) h(xh )+h(yh ) (ii) = h(zh ). 2 Suppose that we have chosen xg , yg and zg for g < h ∈ H. ˜ The homeomorphism h satisﬁes Let Yh = span{{xg , yg , zg | g < h} ∪ D). the assumptions of Lemma 5.5.12, so we can ﬁnd a copy Ch ⊂ E ∩ A of the Cantor set with the properties (a) and (b) of 5.5.12. The linear dimension of the space Yh is less than c, hence we can ﬁnd distinct vectors af , bf ∈ Ch for f ∈ H, such that span{af , bf } ∩ Yh = {0} for every f ∈ H. By the h(af )+h(bf ) property (a) of 5.5.12, the vectors is in the range of h, for every 2 h(a )+h(b ) f f −1 f ∈ H. Let cf = h ( ). Now, we have the following: 2 Claim. There exists f ∈ H such that cf ∈ / span({af , bf } ∪ Yh ). Otherwise, for every f ∈ H, we would have that cf = tf af + sf bf + df , where tf , sf ∈ R and df ∈ Yh . Then df = cf − tf af − sf bf and, by the property (b) of 5.5.12, the set {df | f ∈ H} ⊂ Yh is linearly independent. This contradicts the fact that the linear dimension of the space Yh is less than c. We complete the inductive step of the construction by taking xh = af , yh = bf and zh = cf , for f from the claim. Finally, we put X = span({xh , yh | h ∈ H} ∪ D). Obviously, X is a dense linear subspaces of E. We shall verify that the space X has the required property. Suppose F is a Fr´echet such that (E, F ) does not have

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the property (⋆) and h a homeomorphism of X × lf2 onto a convex subset G of F . By Lavrientiev Theorem, we can extend h to a homeomorphism g between Gδ -subsets of E ⊕ l2 and F , respectively. We can identify F with a closed linear subspace in C(I)ω . Then g ∈ H. Since xg , yg ∈ X, we have g(xg )+g(yg ) g(zg ) = ∈ G, by the convexity of G. Then zg ∈ X ⊕ lf2 which 2 contradicts to the condition (i) of the construction. For a normed space (X, ∥ · ∥), let NX (r) = {x ∈ X : ∥x∥ < r}. For a nonzero functional x∗ ∈ X ∗ and t ∈ R, let H(x∗ , t) is the hyperplane {x ∈ X | x∗ (x) = t}. 5.5.15. Lemma. Let E and F be normed spaces. Let U be an open convex subset of E and let h : U → F be an embedding preserving segments. Then h(U ) is open in aﬀ h(U ). Proof. Fix x ∈ U . We need to check that h(U ) is a neighborhood of h(x) in G = aﬀ h(U ). Using appropriate translations we may assume that x = 0 and h(0) = 0 (then G = span h(U )). Take r > 0 such that NE (r) ⊂ U . Since h is an embedding, there exists an ε > 0 such that for every x ∈ E with ∥x∥ = 2r we have ∥h(x)∥ > ε. From the fact that h preserves segments it follows that for every y ∈ h(U ), y ̸= 0, the intersection of the halﬂine {ty | t ≥ 0} with h(U ) contains ∪ an initial segment of length at least ε. Since h(U ) is convex we have G = n∈N nh(U ). Therefore h(U ) contains the ball NG (ε). 5.5.16. Lemma. Let E be a normed space and F be a reﬂexive Banach space. If there exists a homeomorphic embedding h : U → F preserving segments, for some nonempty open convex subset U of E, then the dual space E ∗ is separable. Proof. Without loss of generality we may assume that U = NE (1) and h(0) = 0. Let G = span h(U ). We may additionally assume that G is dense in F . By Lemma 5.5.15, h(U ) is open in G, hence NG (ε) ⊂ h(U ) for some ε > 0. We can ﬁnd r > 0 such that h(NE (r)) ⊂ NG (ε/2). Put C = h(NE (r)). Clearly, C is an open convex subset of G. Let x∗ be a nonzero functional in E ∗ . We will prove that f = x∗ ◦ −1 h |C : C → R is uniformly continuous. Fix t ∈ x∗ (NE (r)) and put Dt = h(H(x∗ , t) ∩ NE (r)) = f −1 (t) and Yt = aﬀ Dt . Then Dt = Yt ∩ C. Observe that Yt is a closed aﬃne subspace of G of codimension 1. We have h−1 (Yt ∩ NG (ε)) ⊂ H(x∗ , t), therefore for t, t′ ∈ NE (r)), t ̸= t′ , the hyperplanes Yt and Yt′ do not intersect in NG (ε). It follows that dist(Dt , Dt′ ) = inf{∥x − y∥ : x ∈ Dt , y ∈ Dt′ } > 0. Fix δ > 0 and take a sequence t1 < t2 < · · · < tn , ti ∈ x∗ (NE (r)) into intervals of length less than δ/2. Let d = min{dist(Dti , Dti+1 ) | i = 1, . . . , n − 1}. One can easily verify that for x, y ∈ C such that ∥x − y∥ < d

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APPLICATIONS II: CONVEX SETS

we have |f (x) − f (y)| < δ, which shows that f is uniformly continuous. Therefore f can be (uniquely) extended to a continuous function g : K → R, where K is the closure of C in F . For every interval J ⊂ R, f −1 (J) is convex, hence the inverse images of the intervals under g are also convex. It follows that g is continuous with respect to the weak topology w on K. Since F is separable and reﬂexive (K, w) is a metrizable compactum, therefore the Banach space C(K, w) is separable. It remains to observe that the map assigning to each x∗ ∈ E ∗ the function g deﬁned in the above way (we map the zero functional onto the zero function) is an isomorphic embedding of E ∗ into C(K, w). 5.5.17. Corollary. For every Banach space with a nonseparable dual E ∗ and every reﬂexive Banach space F the pair (E, F ) does not have the property (⋆). Proof. Suppose Z is a linear subspace in E such that the closure Z¯ has ﬁnite codimension in E. The the dual space Z¯ ∗ = Z ∗ has ﬁnite codimension in E, and thus is nonseparable. Lemma 5.5.16 completes the proof. Proof of Theorems 5.5.8 and 5.5.9. It follows from 5.5.14 and 5.5.17 that l1 contains a dense linear subspace X such that no convex set in a reﬂexive Banach space is homeomorphic to X × lf2 . Let lf1 = {(ti ) ∈ l1 | ti = 0 for almost all i}. By 5.3.12, the spaces lf1 and lf2 are homeomorphic. Thus X ×lf2 is homeomorphic to Y = X × lf1 . Notice that Y is linearly homeomorphic to a dense linear subspace in l1 . Clearly, Y × σ ∼ = Y , and thus Y is a Zσ -space, and by 5.3.17, Y is an absorbing space. By 5.3.20, Y is not homeomorphic to Y × Y . By similar arguments we can deduce 5.5.9 from 5.5.14 and the following lemma. 5.5.18. Lemma. For every Banach space E the pair (Rω , E) does not have the property (⋆). Proof. Suppose the contrary, there is a linear subspace X ⊂ Rω such that ¯ has ﬁnite codimension in Rω , and for some open convex subset U ⊂ X X there exists a homeomorphic embedding h : U → E preserving segments. Without loss of generality we may assume that 0 ∈ U , h(0) = 0, and h(U ) is bounded in E. Using the geometric structure of Rω , we can ﬁnd a 2-dimensional linear subspace L ⊂ U . Using the fact that h preserves segments, show that e(L) is a 2-dimensional linear subspace of E and thus can not be bounded, a contradiction. C. A Borel pre-Hilbert space, which is not a Zσ -space. 2 Let E = {(xi )∞ os space. i=1 ∈ l | xi is rational for every i} denote the Erd¨

5.5. SOME COUNTEREXAMPLES

185

5.5.19. Theorem. The pre-Hilbert space L = span E has the following properties: (1) (2) (3) (4)

L L L L

is is is is

a Borel space of the ﬁrst Baire category 3 ; not a Zσ -space; linearly homeomorphic to L ⊕ L; strongly universal.

Proof. Evidently, L = span E is dense in l2 . Then to show that E is not complete, it suﬃces to ﬁnd a point x ∈ l2 \L. Given a ﬁnite subset A ⊂ N we identify RA with the subspace {(xi )i∈N ∈ l2 | xi = 0 if i ̸= A} of l2 and by prA : l2 → RA we denote the natural projection, prA : (xi )i∈N 7→ (xi )i∈A . If A = {i} we write pri in place of pr{i} . Write N = ⊔∞ n=1 Nn as a disjoint union of its subsets with |Nn | = n. Fix n ∈ N and notice that for every q1 , . . . , qn−1 ∈ RNn span{q1 , . . . , qn−1 } is at most (n − 1)-dimensional and thus is∪nowhere dense in RNn . Since RNn is a Baire space, there is an xn ∈ RNn \ q1 ,...,qn−1 ∈QNn span{q1 , . . . , qn−1 } such that ∥xn ∥ ≤ 2−n . Let x ∈ l2 be the point deﬁned by the condition: prNn (x) = xn for every n. We claim that x ∈ / span E. Otherwise, x could be written as x = t1 q1 + · · · + tn−1 qn−1 for some n ∈ N, t1 , . . . , tn−1 ∈ R, and q1 , . . . , qn−1 ∈ E. Then xn = prNn (x) = t1 prNn (q1 ) + · · · + tn−1 prNn (qn−1 ), where prNn (q1 ), . . . , prNn (qn−1 ) ∈ QNn , a contradiction with the choice of xn . Let us show that span E is a Borel space. Fix any x ∈ span E and let n be the minimal number such that x can be written as x = t1 x1 + · · · + tn xn for some t1 , . . . , tn ∈ R, x1 , . . . , xn ∈ E. Clearly, in this case, the vectors x1 , . . . , xn must be linearly independent. Moreover, the numbers t1 , . . . , tn must be linearly independent over Q (that is for every q1 , . . . , qn ∈ Q the equality q1 t1 + . . . qn tn = 0 implies q1 = · · · = qn = 0). Otherwise, we can express one of them, to say, tn as tn = q1 t1 + · · · + qn−1 tn−1 , where q1 , . . . , qn−1 ∈ Q. Then x = t1 (x1 + q1 xn ) + · · · + tn−1 (xn−1 + qn−1 xn ), a contradiction with the minimality of n. Furthermore, since the vectors x1 , . . . , xn ∈ l2 are linearly independent, there is a ﬁnite subset A ⊂ N such that |A| = n and the projections prA (x1 ), . . . , prA (xn ) of x1 , . . . , xn onto RA are linearly independent (this can be proved by induction on n). Let qi = prA (xi ), i = 1, . . . , n. Then x ∈ S(A, q1 , . . . , q|A| ), where ∑|A| S(A, q1 , . . . , q|A| ) = { i=1 ti xi | t1 , . . . , t|A| ∈ R are linearly independent over Q, and xi ∈ E ∩ pr−1 A (qi ) for i = 1, . . . , |A|}. Let 3 In

in ℓ2 .

[1999a] Banakh proved that the space span E is a countable-dimensional Fσδ -set

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APPLICATIONS II: CONVEX SETS

A = {(A, q1 , . . . , q|A| ) | A is a ﬁnite subset of N, and (q1 , . . . , q|A| ) ∈ QA are linearly independent vectors in RA }. Evidently, the set A is countable. It follows from the discussion above that ∪ span E = S(A, q1 , . . . , q|A| ). (A,q1 ,...,q|A| )∈A

Now to show that span E is Borel, it suﬃces to show that each set S(A, q1 , . . . , q|A| ) is Borel. Denote by liq(|A|) the set {(t1 , . . . , t|A| ) ∈ RA : t1 , . . . , t|A| are linearly independent over Q}, and remark that S(A, q1 , . . . , q|A| ) is a continuous image of the space P = ∏|A| liq(|A|) × i=1 E ∩ pr−1 A (qi ) under the map χ acting as χ : (t1 , . . . , t|A| , x1 , . . . , x|A| ) 7→

|A| ∑

ti x i .

i=1

Remark that liq(|A|) is a Gδ -set in R|A| and E ∩ pr−1 A (qi ) is an Fσδ subset in l2 for each i, and thus P is a Borel space. According to A.Kechris [1995, 15.1], to show that S(A, q1 , . . . , q|A| ) is Borel, it is enough to verify that χ is injective. Suppose x = (t1 , . . . , t|A| , x1 , . . . , x|A| ), x = (t′1 , . . . , t′|A| , x′1 , . . . , x′|A| ) are two points in P such that χ(x) = χ(x′ ). Then |A| ∑ i=1

ti qi = prA

|A| (∑ i=1

)

′

ti xi = prA ◦χ(x) = prA ◦χ(x ) =

|A| ∑ i=1

t′i prA (x′i )

=

|A| ∑

t′i qi ,

i=1

and thus ti = t′i for i = 1, . . . , |A|, because q1 , . . . , q|A| are linearly independent. To show that xi = x′i for i = 1, . . . , |A|, it is enough to verify that for every j ∈ N prj (xi ) = prj (x′i ). Fix j ∈ N. Then χ(x) = χ(x′ ) ∑|A| ′ and ti = t′i , i = 1, . . . , |A|, imply i=1 ti (prj (xi ) − prj (xi )) = 0. Since t1 , . . . , t|A| are linearly independent over Q, this yields prj (xi ) = prj (x′i ) for every i = 1, . . . , |A|. Thus χ is bijective, and span E is a Borel space. Since it is not complete, by S.Banach [1931], span E is of the ﬁrst Baire category. Now we show that span E is not a Zσ -space. Since span E is homotopy dense in l2 , it suﬃces to verify that the Erd¨os space E is contained in no 2 Zσ -subset of l2 . Suppose Zσ -set ∪∞on the contrary that E ⊂ Z ⊂ l for some 2 Z ⊂ l . Write Z = n=1 Zn , where each Zn is a Z-set in l2 . Further

5.5. SOME COUNTEREXAMPLES

187

for every n ∈ N we identify Rn with the subspace {(xi ) ∈ l2 | xi = 0 for i > n} in l2 . Since Z1 is a Z-set in l2 and E ∩ lf2 is dense in l2 , there are x1 ∈ lf2 ∩ E and ε1 ∈ (0, 1] such that B(x1 , ε1 ) ∩ Z1 = ∅. Find n1 ∈ N with x1 ∈ Rn1 and let B1 = {x ∈ B(x1 , ε1 /2) | pri (x) = pri (x1 ) for all i ≤ n1 }. Clearly B1 is not a Z-set in l2 , so B1 ̸⊂ Z2 and thus we can ﬁnd x2 ∈ B1 ∩ lf2 ∩ E and a positive ε2 ≤ ε1 /2 such that B(x2 , ε2 ) ⊂ B(x1 , ε1 ) and B(x2 , ε2 ) ∩ Z2 = ∅. Find n2 > n1 with x2 ∈ Rn2 and let B2 = {x ∈ B(x2 , ε2 /2) | pri (x) = pri (x2 ) for all i ≤ n2 }. Proceeding in this way we construct sequences (xk ) ⊂ E ∩ lf2 , (εk ) ⊂ (0, 1], and (nk ) ⊂ N such that for every k the following conditions are satisﬁed: B(xk , εk ) ⊂ B(xk−1 , εk−1 ), B(xk , εk )∩Zk = ∅, εk ≤ εk−1 , and pri (xm ) = pri (xk ) for all i = 1, . . . nk and m ≥ k. ∩∞ The the intersection k=1 B(xk , εk ) contains a unique point x. Clearly, x∈ / Z. Furthermore, for every i and every k with i ≤ nk we have pri (x) = pri (xk ) ∈ Q, i.e., x ∈ E ⊂ Z, a contradiction. Exercises to §5.5. 1. Show that every inﬁnite-dimensional normed linear space Y is homeomorphic to a non-open subset of Y . 2. Show that for every inﬁnite-dimensional normed space Y there exists a bijective map g : Y → Y such that g|Y \K is not a homeomorphism for any compact K ⊂ Y . Hint: Use the fact that the unit sphere S of Y is not compact, and thus admits a map λ : S → (0, 1] with inf λ(S) = 0. Deﬁne f : Y → Y letting f (0) = 0 and f (y) = λ(y/∥y∥) · y for y ̸= 0. 3. Let Y be a space from 5.5.2. Show that for all closed linear subspaces E, F ⊂ Y of ﬁnite codimension in Y the following conditions are equivalent: a) E and F are homeomorphic; b) E and F are linearly homeomorphic; and c) codim E = codim F . 4. Suppose X = l2 and Y ⊂ X be a linear subspace satisfying the condition (∗) of 5.5.1. Let L ⊂ Y , dim L = ∞, be a closed linear subspace of inﬁnite codimension in Y . Show that for linear spaces G, F , a homeomorphness L × F ∼ = L × G implies dim F = dim G. 5. (Open Problem) Does there exists a pre-Hilbert Zσ -space with the properties (3)—(6) of 5.5.2? 6. (Open Problem) Is there a Borel locally convex space satisfying the conditions (1)—(4) of 5.5.8 or 5.5.9. ∼ G × σ is an absorbing 7. Prove that l2 contains an additive subgroup G such that G = space not homeomorphic to any convex set in a Fr´ echet space. Below E stands for the Erd¨ os subspace of l2 . 8. (Open Problem) Estimate the Borel type of span E. 9. (Open Problem) Is the space span E countable-dimensional? 10. (Open problem) Is the space span E homeomorphic to span Qω in 11. (Open Problem) Describe the class F0 (span E).

Rω ?

12. Let K be the Cook continuum (see H.Cook [1967] and p.94), and let {Aα }α∈c be a

188

APPLICATIONS II: CONVEX SETS

family of pairwise disjoint subcontinua of K of cardinality c. Supposing that K ω is a linearly independent subset in l2 , let Lα = span(Aω α ) for α ∈ c. a) Show that each Lα is a pre-Hilbert space that is a σ-compact absorbing AR; b) Show that Lα ’s are pairwise nonhomeomorphic, moreover for every distinct α ̸= β there is no continuous surjection Lα → Lβ ; c) Show that in every Borel class Mα \Aα or Aα \Mα there is a continuum many pairwise nonhomeomorphic pre-Hilbert spaces. Hint: see Cauty [1992]. 13. Suppose K is a linearly independent compact subset of l2 . Show that for every zerodimensional subset A ⊂ 2ω , the pre-Hilbert space span(A) is not M1 [1]-universal.

§5.6. Spaces of probability measures It is not the purpose of this section to give a complete exposition of the topology of spaces of (probability) measures; the reader is reﬀered to Fedorchuk [1991]. We will use without proof some results that concern categorial aspects of the measure theory. We begin with the compact case. As usual, the Banach space C(X) of continuous functions on a compactum X is endowed with the sup-norm. A measure on X is, by deﬁnition, a continuous real-valued linear functional on C(X) (i.e. an element of C(X)∗ ). The term “measure” is motivated by the Riesz theorem on isomorphism between the adjoint space C(X)∗ and the space of ﬁnite regular measures on X (see Fedorchuk [1991]). Having in mind this isomorphism, we will also use the term ”measure” for a countablyadditive function deﬁned on the family of all Borel subsets of X. A measure µ is called positively-determined if µ(ϕ) ≥ 0 for every ϕ ≥ 0. A positively-determined measure is called a probability measure if ∥µ∥ = 1. The set P (X) of ∏ all probability measures on X is naturally embeddable into the product {Rϕ | ϕ ∈ C(X)}. We endow P (X) with the topology induced by this embedding (weak∗ topology). Note that the family of sets of the form O(µ, ϕ1 , . . . , ϕk , ε) = {µ′ ∈ P (X) : |µ′ (ϕi ) − µ(ϕi )| < ε, i = 1, . . . , k}, ε > 0, ϕ1 , . . . , ϕk ∈ C(X), is a base of the weak∗ topology on P (X). It is known that P (X) is a metrizable compactum if so is X. For each x ∈ X the Dirac measure δx ∈ P (X) is deﬁned by the condition: δx (ϕ) = ϕ(x), ϕ ∈ C(X). The map δ : X → P (X), x 7→ δx is an embedding. The support supp(µ) of µ ∈ P (X) is deﬁned by the condition: x ∈ / supp(µ) iﬀ there exists a neighborhood U of x such that µ(f ) = 0 whenever f ∈ C(X) and f |(X\U ) ≡ 0. The topological classiﬁcation of the spaces P (X) for compacta X is wellknown. If X is ﬁnite, then P (X) is aﬃnely homeomorphic to (|X| − 1)-

5.6. SPACES OF PROBABILITY MEASURES

189

dimensional simplex. If X is inﬁnite, by Klee theorem 5.1.2, P (X) is homeomorphic to the Hilbert cube Q. For noncompact spaces, we will be interested in the subspaces Pβ (X) and Pˆ (X) of the space P (X). ˇ As usual, βY denotes the Stone-Cech compactiﬁcation of a (separable metrizable) space Y . Let Pβ Y = {µ ∈ P (βY ) | supp(µ) ⊂ Y ⊂ βY }. It is known that we can take any compactiﬁcation cY instead of βY in the above deﬁnition of Pβ (Y ) (see Chigogidze [1984]). As a consequence, we obtain that Pβ (Y ) is separable metrizable if so is Y . In the sequel we drop the index β in the notation Pβ (Y ). Below we give a complete topological classiﬁcation of spaces P (Y ) where Y is a noncompact coanalytic space. 5.6.1. Theorem. Let X be an inﬁnite compactum and Y a proper dense subset. Then the pair (P (X), P (Y )) is homeomorphic to: (i) (Q, σ) iﬀ Y is discrete; (ii) (Q, Σ) iﬀ Y is open and nondiscrete; (iii) (Q, Ω2 ) iﬀ Y is a nonopen Gδ -subset of X; (iv) (Q, Π2 ) iﬀ Y ∈ P2 \ M1 . 5.6.2. Corollary. The space P (Y ) is homeomorphic to: (i) σ iﬀ Y ∼ = N; (ii) Σ iﬀ Y is locally compact nondiscrete noncompact; (iii) Ω2 iﬀ Y is nonlocally compact topologically complete; (iv) Π2 iﬀ Y ∈ P2 \ M1 . In the sequel X denotes an inﬁnite compactum and Y a proper dense subset in X. Clearly P (Y ) is a dense convex subset of P (X), and thus P (Y ) ∈ AR is homotopy dense in P (X). We need some auxiliary results. 5.6.3. Lemma. The set P (Y ) is contained in a σZ-set in P (X). Proof. Let y ∈ X \ Y and U = X \ {y}. Then P (Y ) ⊂ P (U )∪and it ∞ remains to show that P (U ) is a σZ-set in P (X). We have U = i=1 Ki where ∪∞ K1 ⊂ K2 ⊂ . . . is an increasing set of compacta. Then P (U ) = i=1 P (Ki ). Deﬁne a homotopy H : P (X) × [0, 1] → P (X) by the formula H(µ, t) = tδy +(1−t)µ. Then H(µ, 0) = µ for each µ ∈ P (X) and H(P (X)× (0, 1]) ∩ P (Ki ) = ∅ for each i. Thus, each P (Ki ) is a Z-set in P (X). 5.6.4. Lemma. (i) If Y is an open subset in X, then P (Y ) is σ-compact; (ii) if Y ∈ M1 , then P (Y ) ∈ M2 ; (iii) if Y ∈ P2 , then P (Y ) ∈ P2 . Proof. In fact, (i) is proved in Lemma 5.6.3 (one has to take an arbitrary open subset instead of U ).

190

APPLICATIONS II: CONVEX SETS

∩∞ ∪∞ (ii) Let Y = i=1 Un where Un are open in X. We have Un = m=1 Knm 1 where Knm = {x ∈ X∪ | d(x, X \ Un ) ≥ m } (d is any ﬁxed metric in X). ∩∞ ∞ m Then P (Y ) = n=1 m=1 P (Kn ). Since P (Knm ) are compacta, we have P (Y ) ∈ M2 . (iii) Note that P (X) \ P (Y ) = {µ ∈ P (X) | supp(µ) ∩ (X \ Y ) ̸= ∅} = pr1 (E) where pr1 : P (X) × X → P (X) is the projection onto the ﬁrst factor and ∞ ∩ E = (P (X) × (X \ Y )) ∩ ( En ) n=1

where En = {(µ, x) ∈ P (X) × X | B(x, 1/n) ∩ supp(µ) ̸= ∅}, n ∈ N. Since the map supp : P (X) → exp(X) is lower semicontinuous (see, e.g., Fedorchuk, Filippov [1988]), the∩ set En ⊂ P (X)×X is open for every n ∈ N. ∞ We have (P (X) × (X \ Y )) ∩ ( n=1 En ) ∈ P1 and hence P (X) \ P (Y ) = pr1 (E) = P1 , i.e., P (Y ) ∈ P2 . 5.6.5. Lemma. If Y is a nonopen subset of X, then the pair (P (X), P (Y )) is (M0 , M2 )-preuniversal. We will need some auxiliary results. Let G = {(0, 0)} ∪ {( n1 , 0) | n ∈ 1 ) | n, m ∈ N} ⊂ R2 and H = G \ {( n1 , 0) | n ∈ N}. N} ∪ {( n1 , (nm) The proof of the following proposition is elementary and we left it to the reader. 5.6.6. Proposition. If Y is a nonopen subset of X, then X contains a closed subset E such that the pair (E, E ∩ Y ) is homeomorphic to (G, H). 5.6.7. Proposition. If F ∈ M2 (K) for some compactum K, then there exists a map ξ : K → P (G) such that ξ −1 (P (H)) = F . 1 Proof. For n ≥ 1 let Gn = {( n1 , 0)} ∪ {( n1 , (nm) ) | m ≥ 1} and Hn = ∩∞ 1 Gn \ {( n , 0)} = Gn ∩ H. Let F = n=1 Fn , where Fn , n ≥ 1 are Fσ -sets in K. Fix n ≥ 1 and construct a map ξn : K → P (G∪n ) such that ξ −1 (P (Hn )) = ∞ 1 Fn . For m ≥ 1 put αm = ( n1 , (nm) ). Let Fn = m=1 Lm where L1 ⊂ L2 ⊂ · · · is a sequence of compacta in K. Deﬁne the map ξn by the formula

ξn (x) =

∞ ∑

2−m d(x, Lm )δαm+1 + (1 −

m=1

(d is a metric on K restricted by 1).

∞ ∑ m=1

2−m d(x, Lm )δα1 )

5.6. SPACES OF PROBABILITY MEASURES

191

Now∑ the required map ξ : K → P (G) can be deﬁned by the formula ∞ ξ(x) = n=1 2−n ξn (x). The proof of Lemma 5.6.5 is an immediate consequence of Propositions 5.6.6 and 5.6.7. 5.6.8. Lemma. If Y ∈ / M1 is a coanalytic subset of X, then the pair (P (X), P (Y )) is (M0 , P2 )-preuniversal. Proof. Fix a pair (K, C) ∈ (M0 , P2 ). Since K\C is an analytic set, there is a Gδ -subset G ⊂ K × 2ω such that prK (G) = K\C. It follows from Milutin Lemma (see Fedorchuk, Filippov [1988]) that there is a map ξ : K0 → K × 2ω of a zero-dimensional compactum K0 onto K × 2ω such that the map P (ξ) : P (K0 ) → P (K × 2ω ) has a section s : P (K × 2ω ) → P (K0 ). By Hurewicz Theorem 2.1.4, the pair (X, Y ) is (M0 [0], A1 )-preuniversal, thus there is a map η : K0 → X with η −1 (Y ) = K0 \ξ −1 ((K × 2ω )\G). Fix any measure µ0 ∈ P (2ω ) with supp(µ0 ) = 2ω . Now deﬁne the map f : K → P (X) letting f (x) = P (η) ◦ s ◦ (δx ⊗ µ0 ), x ∈ K. The reader can verify that f −1 (P (Y )) = C. Proof of Theorem 5.6.1. The statements (i) and (ii) follow from 5.3.12, 5.3.15, and 3.1.7. (iii) By Lemma 5.6.3, P (Y ) is contained in a σZ-set in P (X), and by Lemma 5.6.4, P (Y ) ∈ M2 . Using Lemma 5.6.5 and 3.2.11, we conclude that the pair (P (X), P (Y )) is (M0 , M2 )-universal, and thus the space P (Y ) is M2 -universal. Since P (Y ) is closed in its aﬃne hull, by 5.2.7 and 5.3.6, P (Y ) is an M2 -absorbing AR, homeomorphic to Ω2 = Σω . By 3.1.10 the pair (P (X), P (Y )) is homeomorphic to (Q, Ω2 ). The proof of (iv) is analogous. We have only to use Lemma 5.6.8 instead of Lemma 5.6.5. Now we consider the spaces of Radon measures. Let Y be a subspace of a compactum X. Put Pˆ (Y ) = {µ ∈ P (X) | µ∗ (Y ) = 1} ⊂ P (X), where µ∗ (Y ) = sup{µ(B) | B is a Borel subset of Y }. It is well-known that the topological type of the space Pˆ (Y ) does not depend on embedding Y into a compactum X, so we will use the notation Pˆ (Y ) not only for subspaces of compacta but also in the absolute case. There exists an instrict description of the space Pˆ (Y ). A probability measure µ is called a Radon measure if for every Borel subset B ⊂ Y and every ε > 0 there exists a compactum K ⊂ B such that µ(B \ K) < ε. It turns out that Pˆ (Y ) is exactly the space of the Radon measures on Y . Note that Pˆ (Y ) is a convex dense subset in P (X) and hence Pˆ (Y ) ∈ AR is homotopy dense in P (X).

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In the sequel we will use the functoriality of Pˆ . For a map f : X → Y we deﬁne the map Pˆ (f ) : Pˆ (X) → Pˆ (Y ) by the formula: Pˆ (f )(µ)(A) = µ(f −1 (A)) where µ ∈ Pˆ (X) and A is a Borel subset of Y . Note that Pˆ preserves closed embeddings and preimages, i.e., Pˆ (f )−1 (Pˆ (M )) = Pˆ (f −1 (M )) where M ⊂Y. We will give a complete topological classiﬁcation of the spaces Pˆ (Y ) for Borel spaces Y . For this we will investigate topology of the pairs (P (X), Pˆ (Y )) where Y is a dense Borel subset of an inﬁnite compactum X. 5.6.9. Proposition. Let Y ∈ Mξ (X) where ξ is a countable ordinal. Then for every a ∈ [0, 1] we have {µ ∈ P (X) | µ(Y ) ≥ a} ∈ Mξ (P (X)). Proof. We can immediately check that the set {µ ∈ P (X) | µ(Y ) ≥ a} is a closed subset of P (X) whenever Y is closed in X. Thus the proposition is valid for ξ = 0. Assume that the proposition is already proved for every ξ < β. Let ∩∞ Y ∈ Mβ (X) and a ∈ [0, 1]. Then Y = i=1 Ai where∩Ai ∈ Aξi (X) for ∞ {µ ∈ P (X) | some ξi < β. ∩We have {µ ∈ P (X) | µ(Y ) ≥ a} =∩ i=1 ∞ ∩∞ ∞ µ(Ai ) ≥ a} = i=1 {µ ∈ P (X) | µ(X \ Ai ) ≥ 1 − a} = i=1 j=1 Aij where Aij = {µ ∈ P (X) | µ(X \ Ai ) < 1 − a + 1j }. Since X\Ai ∈ Mξi (X), by inductive assumption, P (X)\Aij ∈ Mξi (P (X)) for every j ∈ N. Then Aij ∈ Aξi (X) and {µ ∈ P (X) | µ(Y ) ≥ a} ∈ Mβ (P (X)). 5.6.10. Corollary. Y ∈ Mξ (X) iﬀ Pˆ (Y ) ∈ Mξ (P (X)). Proof. Necessity can be obtained from Proposition 5.6.9; take a = 1. Suﬃciency easily follows from existence of the closed embedding δ : Y → Pˆ (Y ). Recall that X is a Baire space ∩∞ if for every sequence {Ui } of open dense subsets in X the intersection n=1 Un is dense in X. 5.6.11. Proposition. If Y ⊂ X is not a Baire space, then Pˆ (Y ) is contained in a σZ-subset of P (X). Proof. There exists a nonempty open subset U of Y of the ﬁrst Baire category. Let F = ClX (Y \U ). There exists a sequence ∪∞F1 ⊂ F2 ⊂ . . . of closed nowhere dense subsets in X such that U ⊂ n=1 Fn . Obviously, Y ⊂ F ∪ (∪∞ n=1 Fn ). Then (∗)

Pˆ (Y ) = Pˆ (F ) ∪ (∪∞ n=1 An )

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where An = {µ ∈ Pˆ (X) | µ(Fn ) ≥ 2−n }, n ≥ 1. Since F ̸= X, we have that Pˆ (F ) is a Z-set in P (X). Indeed, the set Pˆ (F ) is closed in P (X) and Pˆ (X)\Pˆ (F ) is a convex dense subset in the AR-space P (X). Thus the set Pˆ (F ) is homotopy negligible in P (X) and hence is a Z-set in P (X). Similarly, the sets An , n ∈ N are Z-sets in P (X) and, by (∗), we are done. 5.6.12. Proposition. Let Y be a dense Borel subset of X. If Y is a Baire space, then P (X)\Pˆ (Y ) is contained in a σZ-subset of P (X). Proof. Since Y is a Borel subset of X, we have Y = G ∪ F where G is a Gδ subset and F is a ﬁrst category set in X. Then P (X)\Pˆ (Y ) ⊂ P (X)\Pˆ (G). Since ∪∞ the space Y is a Baire space, the set G is dense in X. Then X\G = of closed nowhere dense subsets n=1 Fn where F1 ⊂ F2 ⊂ . . . is a sequence ∪∞ in X. Therefore, P (X)\Pˆ (G) = n=1 An , where An = {µ ∈ P (X) | µ(Fn ) ≥ 2−n }, n ∈ N. It is easy to see that each An is a Z-set in P (X). ∪∞ Then P (X)\Pˆ (Y ) is contained in a σZ-set P (X)\Pˆ (G) = n=1 An . Now we can pass to the problem of topological classiﬁcation of the pairs (P (X), Pˆ (Y )). In the case Y ∈ M1 , we can apply Theorem 5.2.8 to prove the following theorem. 5.6.13. Theorem. If Y is a proper dense Gδ -subset of X, then the pair (P (X), Pˆ (Y )) is homeomorphic to the pair (Q, s). We will need some auxiliary constructions. Let C = 2ω denote the Cantor cube. We construct the pairs (C, Aα (C)), (C, Mα (C)), α being a countable ordinal. Fix ∗ ∈ C and let (C, M0 (C)) = (C, {∗}) and (C, A0 (C)) = (C, C\{∗}). Assume that for every ξ < α the pairs (C, Mξ (C)) and (C, Aξ (C)) are already con∼ (C ω , Aα−1 (C)ω ), if α is a nonlimit ordistructed. Let (C, M∏ α (C)) = ∏ ∼ nal, (C, Mα (C)) = ( ξ