Spaces with fragmentable open sets

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Abstract

The object of the paper are the regular topological spaces X in which there exists a metric d related to the topology in the following way: for every nonempty open subset U of X and for every ε>0 there exists a nonempty open subset V of U with d-diameter less than ε. It is shown that such a space X is pseudo-almost Čech complete if, and only if, it contains a dense completely metrizable subset (X is pseudo-almost Čech complete if it is a subset of some pseudocompact space Y and contains as a dense subset some Gδ-subset of Y).

A large class of spaces L is described (containing all Borel subsets of compact spaces, all pseudocompact spaces and all p-spaces of Arhangel'skii) such that, if X belongs to L and admits a metric d with the above property, then X is “metrizable up to a first Baire category subset”. I.e. X is the union of two sets X1 and X2 where X1 is of the first Baire category in X and X2 is metrizable. We provide also different characterizations of the class L.

Introduction

We continue our study of regular topological spaces (X,τ) 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 ε>0 there is a nonempty open set V such that VU is not empty and the d-diameter of VU 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 (X,τ) is the union of two disjoint sets X1 and X2, where X1 is of the first Baire category in X and X2 (if not empty) admits a continuous one-to-one mapping into a metrizable space. It is easy to realize that X2 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 (X,τ) for which there is a metric d fragmenting the open sets and generating a topology finer than τ (i.e. the identity mapping (X,d)(X,τ) 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 (X,τ) the set X2 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 Gδ-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 (X,τ) from this class the existence of a metric d fragmenting the open sets implies that there exists, in general another, metric d 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 L of spaces X which are “conveniently located” (see Definition 3.1) in some feebly compact space Y. The class L 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 L. 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 (X,τ) for which there exists a complete metric d which fragments the open sets of (X,τ) 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), (X,τ) is such a space if, and only if, the set X2X is a dense completely metrizable subset of (X,τ) (necessarily, X2 is also a Gδ - subset of (X,τ)). It was proved in [4] (Corollary 3.3) that, if a compact space (X,τ) 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 (X,τ) 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 Gδ subset of Y. In particular, an almost Čech complete space (X,τ) 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 (T1 and T3 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 A.

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]) (X,τ) is a fos-space if and only if there exist families (γn)n1 of disjoint nonempty open sets in X such that:

  • (1)

    for every n1 the open set Wn:={Vn:Vnγn} is dense in X;

  • (2)

    for every Vn+1γn+1 there exists some (necessarily uniquely determined) set Vnγn such that Vn+1Vn;

  • (3)

    if (Vn)n1 is a sequence of sets such that Vnγn for every n1 and (Vn)n1 is nested (i.e. Vn+1Vn for each n1), then the intersection nVn is either

A class L of spaces for which fos-property and strong fos-property coincide

The class L 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 U1 of Z such that U1X. Player II answers by choosing an open set V1 of Z such that V1U1 and V1X. 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).

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