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Absorbing Sets in Infinite-Dimensional Manifolds 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 charac- terization 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´nczyktheo- rems was realized in the fundamental paper of M.Bestvina and J.Mogilski. Starting from Anderson's cap-sets, Bessaga-Pe lczy´nski'sskeletoids, and ab- sorbing 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 charac- terization theorems can be proved. Besides, they possess many remarkable topological properties, in particular, their topological and homotopy classi- fications 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 mani- folds. 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 condi- tions. 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 prod- ucts, hyperspaces, topological groups) and in the fifth chapter (convexity and related topics). Lviv, July 1996 4 Contents1 Preface 3 Chapter I: Basic Theory 6 1.1. Preliminaries 5 1.2. Homotopy dense and homotopy negligible sets 13 1.3. The strong discrete approximation property 17 1.4. Z-Sets and strong Z-sets 23 1.5. Strong universality 29 1.6. Absorbing and coabsorbing spaces 38 1.7. Strongly universal and absorbing pairs 44 Chapter II: Constructions of absorbing spaces 53 2.1. Preliminaries I: Descriptive Set Theory 53 2.2. Preliminaries II: Dimension Theory 56 2.3. Invertible and soft maps 69 2.4. Absorbing spaces for [0; 1]-stable classes 81 2.5. Weak inverse limits and absorbing sets 86 2.6. Some negative results 92 Chapter III: Advanced Topics 95 3.1. Interplay between strongly universal spaces and pairs 95 3.2. Characterizing (strong) C-universality for \nice" classes C 106 Chapter IV: Applications I. 125 4.1. Infinite products 125 4.2. Topological groups 128 4.3. Hyperspaces 137 Chapter V: Applications II: Convex Sets 147 5.1. Locally compact convex sets 147 5.2. Topologically complete convex sets 150 5.3. Strong universality in convex sets 154 5.4. Strong universality in locally convex spaces 166 5.5. Some counterexamples 172 5.6. Spaces of probability measures 188 ∗ 5.7. Function spaces Cp(X) and Cp (X) 196 References 199 1The 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 nec- essary material in Sections 1.1|1.5. Section 1.7 contains Uniqueness Theo- rem 1.7.6 for absorbing pairs and Theorem 1.7.9 revealing interplay between strongly universal spaces and pairs. x1.1. 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¨ofand 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], Ik the k-dimensional cube, and @Ik 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]!: We will consider also the following subspaces of Q: the pseudo-interior s = (−1; 1)!, the pseudo-boundary Bd(Q) = Qns, f 2 j j j g the radial interior Σ = (ti) Q supi ti < 1 of the Hilbert cube, and subspaces Qf = f(ti) 2 Q j ti =6 0 for finitely many of ig and σ = Qf \ s: The standard separable Hilbert space is denoted by l2 and 2 f 2 2 j 6 g lf = (ti) l 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, Bd(A) is the boundary of A, i.e. Bd(A) = A¯n Int(A). Finally, B(A; ") = fy 2 X j d(A; y) < "g and D(A; ") = fy 2 X j d(A; y) ≤ "g denote the open and closed balls about A with radius ", respectively. If A = fxg for some x then we write B(x; ") instead of B(fxg;") and D(x; ") instead of D(fxg;"), respectively. If f : X ! Y is a function and A ⊂ X then fjA 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) < 1. Noticing that minf1; dg 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 comple- te-metrizable separable topological space. A Baire space, by definition, is f g a space X such that for every countableT collection Gn n2N consisting of M open dense subsets of X, the intersection n2N Gn is dense in X. By 0, 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 = ft · a j t 2 F; a 2 Ag and A + B = fa + b j a 2 A; b 2 Bg. It will be convenient to introduce the following notation. For U, V collec- tions 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 2 U there is V 2 V with U ⊂ V . Let X be a space, A ⊂ X, and U a collection of subsets of X. Let U\ A = fU \ A j U 2 Ug. The star of A with respect to U is the set St (A; U) = [fU 2 U j A \ U =6 ;g: The collection fSt (U; U) j U 2 Ug is denoted by St (U). For every n ≥ 0 de- fine St n(U) inductively letting St 0(U) = U and St n(U) = St (St n−1(U)). We set mesh U = supfdiam(U) j U 2 Ug: For a map f : Y ! X let f −1U = ff −1(U) j U 2 Ug. Recall that a collection U of subsets of X is called locally finite in X (resp. discrete in X) if each point x 2 X has a neighborhood that meets only finitely many of elements of U (resp. at most one element of U). Let " > 0. We say that two maps f; g : Y ! X are "-close if d(f(y); g(y)) < " for every y 2 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 2 Y with f(y) =6 g(y) there is U 2 U such that f(y); g(y) 2 U. Sometimes this notion will be used in a more general situation: given two maps f : Y ! X and f 0 : Y 0 ! X (with distinct domains) we write (f; f 0) ≺ U if the restrictions fjY \ Y 0 and f 0jY \ Y 0 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 2 cov(X) there exists a map g : Y ! X which possesses that property and is U-close to f.
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