The infinite-dimensional topology of function spaces (Q2721306)

For a completely regular space \(X\), let \(C_p(X)\) be the space of all continuous real-valued functions on \(X\) with the topology of pointwise convergence. The first aim of this book is to introduce the reader to some recent results on the topology of these spaces \(C_p(X)\), in particular to the theorem of Dobrowolski-Marciszewski-Mogilski asserting that, if \(X\) is countable and if \(C_p(X)\) is an \(F_{\sigma\delta}\) in \(\mathbb{R}^X\), then \(C_p(X)\) is homeomorphic to the countable product \(B(Q)^\infty\), where \(B(Q)\) is the pseudo-boundary of the Hilbert cube \(Q\). The proofs of these results use a great variety of topological techniques and, to make them accessible to a student with limited background, the first five chapters of the book are devoted to the development of the necessary techniques. As these techniques have many applications, the exposition goes beyond the needs of \(C_p\)-theory, which makes the book useful for a larger audience.NEWLINENEWLINENEWLINEChapter 1 presents selected basic facts on the topology of separable metrizable spaces: linear metric spaces, function spaces, basic results on ANRs, topological characterization of some simple spaces (Cantor set, irrationals, arc), inductive constructions of homeomorphisms, Bing's shrinking criterion, inverse limits, hyperspaces. Applications and examples systematically illustrate these facts.NEWLINENEWLINENEWLINEChapter 2 defines countable simplicial complexes and nerves of countable coverings. The Brouwer fixed point theorem and some of its equivalents are deduced from Sperner's lemma.NEWLINENEWLINENEWLINEChapter 3 is a very nice exposition of dimension theory of separable metrizable spaces. Together with the classical core of the theory, it contains important recent results. Among the topics treated, we mention the following: the dimension of almost zero-dimensional spaces and their products, a detailed study of weakly \(n\)-dimensional spaces, the comparison of the dimension with the earlier Dimensionsgrad of Brouwer, constructions of various types of infinite-dimensional compacta, the coloring of maps and a simple construction of high-dimensional hereditarily indecomposable continua.NEWLINENEWLINENEWLINEChapter 4 treats some classical properties and characterizations of separable ANRs, but most of them are not formulated for spaces, but for a special type of pairs. There are the pairs \((X,Y)\) where \(X\) is an ANR and \(Y\) a subset of \(X\) such that there exists a homotopy \(h: X\times [0,1]\to X\) verifying \(h_0= \text{id}\) and \(h(X\times]0,1])\subset Y\). In the sequel, we will say that such a subspace \(Y\) is homotopically dense in \(X\) (the author calls such a pair an ANR-pair, but this terminology is unfortunate because it has already been used by other authors to designate more general pairs for which it seems better appropriate).NEWLINENEWLINENEWLINEChapter 5 describes some techniques of infinite-dimensional topology. The \(Z\)-sets are introduced, the theorems of extension of homeomorphisms between \(Z\)-sets in \(Q\) and of approximation of maps of compacta by \(Z\)-embeddings are proved. Then capsets are introduced, their principal properties proved and some classical examples of capsets are given. Finally, the author introduces the concepts of absorbing systems and \({\mathcal M}\)-absorbers. \({\mathcal M}\)-absorbers are a variation of the notion of absorbing sets introduced by \textit{M. Bestvina} and \textit{J. Mogilski} [Mich. Math. J. 33, 291-313 (1986; Zbl 0629.54011)]; see the discussion below for the relations between them. It is shown that \(B(Q)^\infty\) is an \({\mathcal F}_{\sigma\delta}\)-absorber, and some other examples of \({\mathcal F}_{\sigma\delta}\)-absorbers are given.NEWLINENEWLINENEWLINEChapter 6 is devoted to \(C_p\)-theory. The problems considered fall into two categories: what can be said about spaces \(X\) and \(Y\) when \(C_p(X)\) and \(C_p(Y)\) are isomorphic as topological vector spaces (\(X\) and \(Y\) are \(\ell\)-equivalent) or when \(C_p(X)\) and \(C_p(Y)\) are homeomorphic (\(X\) and \(Y\) are \(t\)-equivalent). There exist numerous results on \(\ell\)-equivalence, and this chapter contains only a few of them, among which the invariance of \(\sigma\)-compactness and dimension under \(\ell\)-equivalence and, for metrizable spaces, the invariance of complete metrizability. Concerning purely topological properties of \(C_p(X)\), it is shown that \(C_p(X)\) is not a \(G_{\delta\sigma}\) in \(\mathbb{R}^X\) if \(X\) is not discrete. The Baire property of \(C_p(X)\) is characterized; in case \(X\) is countable and has only one nonisolated point, this characterization is translated into set-theoretical properties of the filter, of neighborhoods of the exceptional point. It is shown that if metrizable spaces \(X\) and \(Y\) are \(t\)-equivalent, then \(X\) is a countable union of \(G_\delta\)-subsets which are homeomorphic to \(G_\delta\)-subsets of \(Y\). The author proves that if \(X\) is countable and if \(C_p(X)\) is an \(F_{\sigma\delta}\) of \(\mathbb{R}^X\) (for example if \(X\) is metrizable), then \(C_p(X)\) is an \({\mathcal F}_{\sigma\delta}\)-absorber in the Hilbert cube \(\widehat{\mathbb{R}}^X\) (\(\widehat{\mathbb{R}}= [-\infty,\infty]\)). The author dedicates also a section to the study of the subspace \(C^*_p(X)\) of bounded continuous functions. This chapter is illustrated by many examples, with or without proofs.NEWLINENEWLINENEWLINEThe book contains three appendices. Appendix A collects the topological facts needed in the book. Each section of the book is completed by exercises, and appendix B gives the answers to selected exercises. Appendix C contains historical notes and comments.NEWLINENEWLINENEWLINESome comments about absorbers and absorbing sets are in order. The author proves that, for certain spaces, \(C_p(X)\) is an \({\mathcal F}_{\sigma\delta}\)-absorber in \(\widehat{\mathbb{R}}^X\) and claims that this result is slightly stronger than the theorem of Dobrowolski, Marciszewski and Mogilski asserting that \(C_p(X)\) is homeomorphic to \(B(Q)^\infty\). This is illusory because there exists, up to homeomorphism, a unique pair \((Q,Y)\) where \(Y\) is homeomorphic to \(B(Q)^\infty\) and homotopically dense in \(Q\) [see \textit{T. Banakh} and \textit{R. Cauty}, Interplay between strongly universal spaces and pairs, Diss. Math. 386 (2000; Zbl 0954.57007)]. In contrast, it is interesting to note that, when \(C_p(X)\) is homeomorphic to \(B(Q)^\infty\), there are two possibilities for the pair \((\mathbb{R}^X, C_p(X))\): it is homeomorphic to \((\mathbb{R}^\infty, c_0)\) if \(X\) is compact and to \(((\mathbb{R}^\infty)^\infty, \sigma^\infty)\) if \(X\) is not compact (\(c_0\) is the space of sequences converging to zero, and \(\sigma\) is the subset of almost zero sequences). By their very definition, the absorbers used in this book are homotopically dense subsets of \(Q\), whereas absorbing sets are spaces admitting intrinsic characterizations independent of any compactification or completion, a fact useful when one wants to study a concrete example which is not given with a nice natural compactification homeomorphic to \(Q\). Absorbers and absorbing sets are among the systems that can be characterized by properties of strong universality, as are the pairs \((\mathbb{R}^\infty, c_0)\) and \(((\mathbb{R}^\infty)^\infty, \sigma^\infty)\) mentioned above. The reader who wants to convince himself of the strength of this technique and of the variety of its applications may consult the survey paper of the reviewer [Proc. Steklov Inst. Math. 212, 89-114 (1996); translation from Tr. Mat. Inst. Steklova 212, 95-122 (1996; Zbl 0874.57024)]. The relations between the absorbers of this book and the absorbing sets are simple: for a class \({\mathcal M}\), an \({\mathcal M}\)-absorber is a copy \(Y\) of an \({\mathcal M}\)-absorbing set in \(Q\) such that the pair \((Q,Y)\) is strongly \(({\mathcal K},{\mathcal M})\)-universal, where \(({\mathcal K},{\mathcal M})\) is the class of all pairs \((C,M)\) where \(C\) is compact and \(M\) belongs to \({\mathcal M}\). For some classes \({\mathcal M}\), all copies of an \({\mathcal M}\)-absorbing set homotopically dense in \(Q\) are \({\mathcal M}\)-absorbers, but there are (nonpathological) classes \({\mathcal M}\) and homotopically dense copies \(Z\) of an \({\mathcal M}\)-absorbing set in \(Q\) such that \((Q,Z)\) is not strongly \(({\mathcal K},{\mathcal M})\)-universal, hence it is not an \({\mathcal M}\)-absorber, but such that also \((Q,Z)\) can be characterized by properties of strong universality for a conveniently choosen class of pairs [see Banakh, Cauty, op. cit.]. Thus, if a reader of this book wants to pursue the study of infinite-dimensional topology, he will need to familiarize himself with these more powerful techniques, and we recommend him to read the survey paper cited above and the book of \textit{T. Banakh}, \textit{T. Radul} and \textit{M. Zarichnyi} [Absorbing sets in infinite-dimensional manifolds, VNTL, Lvov, 1996].NEWLINENEWLINENEWLINEThis book is a very good introduction to modern research in some areas of topology, especially dimension theory and \(C_p\)-theory, and we strongly recommend it to students interested in these topics.