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Compact \(G_{\delta}\) sets. (English) Zbl 1101.54034

A Hausdorff space \(X\) is said to be a c-semistratifiable space (or simply CSS-space) if for each compact set \(C\subseteq X\) there are open sets \(G(n,C)\) in \(X\) such that (i) \(\bigcap\{G(n,C):n\in\mathbb N\}=C\), (ii) \(G(n+1,C)\subseteq G(n,C)\) for each \(n\in\mathbb N\) and (iii) if \(C,D\) are compact and \(C\subseteq D\), then \(G(n,C)\subseteq G(n,D)\).
After a short introduction, Section 2 of this paper is dedicated to the class of (Hausdorff) spaces in which every compact subspace is a \(G_\delta\) set and to comparing properties of this class with properties of the class of \(CSS\)-spaces. Among the most important results of this section, it is shown that if \(X\) has a quasi-\(G_\delta\)-diagonal, then any countably compact subset of \(X\) is a compact metrizable \(G_\delta\)-subset of \(X\) and that if \(X\) is regular and has a base of countable order then every compact subspace of \(X\) is a \(G_\delta\)-subset of \(X\). It is also shown here that if \(\mathcal U\) is an open cover of \(X\) such that for each \(U\in \mathcal U\), each compact subset of \(U\) is a \(G_\delta\)-subset of \(U\), then each compact subset of \(X\) is a \(G_\delta\)-subset of \(X\). This last result does not carry over to the class of \(CSS\)-spaces. Section 3 is dedicated to the study of general properties of \(CSS\)-spaces and a characterization of \(CSS\)-spaces is obtained in terms of a sequence \(\{g(n,x)\}\) of open subsets of \(X\). Section 4 studies \(GO\)-spaces which are \(CSS\)-spaces. Among other results, it is shown that any monotonically normal \(CSS\)-space is hereditarily paracompact and (the main theorem of this section) that in the class of \(GO\)-spaces with a dense subset which is the countable union of closed and discrete subsets, being \(CSS\) is equivalent to having a \(G_\delta\)-diagonal.

MSC:

54E99 Topological spaces with richer structures
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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