Spaces with fragmentable open sets☆
Introduction
We continue our study of regular topological spaces in which there is a metric d which is related to the topology τ of the space in the following way: for every nonempty open set U of X and for every there is a nonempty open set V such that is not empty and the d-diameter of is less or equal to ε. Following [4] we call every such space a space with fragmentable open sets or a fos-space, for short. We use also the words the metric d fragments the open subsets of X. The roots of this notion go back to the papers of Thielman [8] and Bledsoe [5] (see [4] for more information). The term “fragmentability of a space” was introduced by Jayne and Rogers in [7] and was used by them for a stronger property: there exists a metric d in X which fragments not only the open subsets U but all nonempty subsets A of X. The class of fragmentable spaces of Jayne and Rogers has established itself as a very good tool in Banach space theory and Convex analysis. Fos-spaces constitute a larger class. They are the object of interest in this paper. The structure of a fos-space is rather simple. As shown in [4] (Theorem 2.4), every fos-space is the union of two disjoint sets and , where is of the first Baire category in X and (if not empty) admits a continuous one-to-one mapping into a metrizable space. It is easy to realize that in this case is fragmentable in the sense of Jayne and Rogers. This allows to say that every fos-space becomes fragmentable in the sense of Jayne and Rogers after removing from it a first Baire category subset. Of special interest for us in this paper are the spaces for which there is a metric d fragmenting the open sets and generating a topology finer than τ (i.e. the identity mapping is continuous). Such spaces will be called here strongly fos-spaces. Our interest to strongly fos-spaces stems from the fact that for every such space the set with the inherited topology is metrizable (by the metric d which fragments the open sets and generates a finer topology ([4], Remark 2.5)). In other words, using loose language, every strongly fos-space is metrizable up to a first Baire category subset. In particular, every Baire and strongly fos-space contains a dense metrizable -subset.
One of the goals of this paper is to determine a class of topological spaces, as large as possible, for which fos-property and strong fos-property coincide. I.e. for spaces from this class the existence of a metric d fragmenting the open sets implies that there exists, in general another, metric which also fragments the open sets and, in addition, generates in X a topology finer than τ. A natural candidate for such a class is the class of spaces X which are “conveniently located” (see Definition 3.1) in some feebly compact space Y. The class contains all Borel subsets of compact spaces, all pseudo-compact spaces and all feathered spaces (p-spaces of Arhangel'skii). Two types of characterizations are given for the spaces from the class . One is via a topological game similar to the famous Banach-Mazur game and the other via the existence in the space Y of a sieve-like structure closely related to X.
Of interests are also the spaces for which there exists a complete metric d which fragments the open sets of and generates in X a topology which is finer than τ. We call these spaces almost completely metrizable or acm-spaces because, as shown in [4] (Theorem 2.6), is such a space if, and only if, the set is a dense completely metrizable subset of (necessarily, is also a - subset of ). It was proved in [4] (Corollary 3.3) that, if a compact space is a fos-space, then there exists a complete metric which fragments the open sets and generates a topology finer than τ. In particular, a compact space is a fos-space if and only if it is a acm-space. I.e. it contains a dense completely metrizable subset. We extend the validity of this statement from compact spaces to subspaces X of pseudocompact spaces Y such that X contains as a dense subset some subset of Y. In particular, an almost Čech complete space is a fos-space if, and only if, it is an acm-space (Corollary 2.5).
The topological spaces we consider in this paper are assumed to be regular ( and separation axioms hold). If stronger separation properties are needed (like complete regularity), this will be stated explicitly. As usual, the closure of a set A will be denoted by .
Section snippets
Characterization of fos-spaces, strongly fos-spaces and acm-spaces
We base our considerations on the following statement.
Proposition 2.1 (Proposition 2.1 in [4]) is a fos-space if and only if there exist families of disjoint nonempty open sets in X such that: for every the open set is dense in X; for every there exists some (necessarily uniquely determined) set such that ; if is a sequence of sets such that for every and is nested (i.e. for each ), then the intersection is either
A class of spaces for which fos-property and strong fos-property coincide
The class of spaces we have in mind will be defined by means of a topological game similar to the classical Banach-Mazur game. Let X be a nonempty subspace of the topological space Z. Two players (Player I and Player II) play a game in Z by selecting open subsets of Z having nonempty intersection with X. More precisely, Player I starts the game by choosing some open set of Z such that . Player II answers by choosing an open set of Z such that and . On the n-th stage of
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The second and third author have been partially supported by the Bulgarian National Fund for Scientific Research (grant DFNI-DNTS/Russia 01/9/2017).