A class of infinite-dimensional topological spaces with applications to the theory of retracts and selection theory

Abstract

In this paper we consider a class of infinite-dimensional topological spaces called C-spaces. We establish a dimension theory for C-spaces and then apply this theory to obtain results concerning the existence of extensions of certain maps. We conclude by giving a selection theorem. In chapter II the class of strongly completely normal (SCN) spaces is defined. It is shown that a space X is SCN if and only if it is hereditarily collectionwise normal. Examples are given delineating the nature of SCN spaces. It is within this class of spaces that most of the theory of C-spaces is developed and applied in the succeeding chapters. Definition: A space X has property C (is a C-space) if for every sequence {C_i} from i=1 to infinity of open covers of X there is a sequence {U} from i=1 to infinity of pairwise disjoint families of open sets such that (1) for each i, if U in U_i then U subset G for some G in C_i, and (2) Union from i=1 to infinity of U_i is a cover of X. Theorem: An F_sigma subspace of an SCN C-space is a C-space. Theorem: In an SCN paracompact space, the union of a locally countable family of C-spaces is a C-space. Theorem: A metric space X is a C-space if and only if for each pair A, B of disjoint closed sets there is an open set U such that A subset U subset X-B and Bd U is a C-space. The classes of countable-dimensional spaces and weakly infinite-dimensional spaces are defined. We show that in metric spaces the following implications hold: Countable dimensionality implies Property C implies weak infinite Dimensionality. Alexandroff's Problem asks whether a weakly infinite-dimensional compact metric space must be countable-dimensional. We discuss this problem as well as consider the products of C-spaces and the images of C-spaces under certain mappings. The principal application of this theory is the following Theorem: Let X and Y be metrizable, A a closed subset of X, and f:A to Y a map. If (1) Y is locally contractible, and (2) Bd A is a C-space, then there exists a continuous extension of f to a neighborhood U of A. If Y is also contractible, we may take U = X. Corollary: For a metrizable C-space X the following statements are equivalent. 1. X is locally contractible. 2. X is an ANR(m). In Chapter V we give sufficient conditions for the existence of a global selection for continuous carriers defined on certain C-spaces. Open questions are pointed out throughout the paper.