## 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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### References:

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