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Relatively compact spaces and separation properties. (English) Zbl 0851.54024

Call a subset \(A\) of a space \(X\) relatively compact if every open cover of \(X\) has a finite subfamily that covers \(A\). A space is potentially compact if it can be embedded in some other space as a relatively compact subset. The authors note that, among Hausdorff spaces, complete regularity implies potential compactness which in turn implies regularity. Using various inputs to Jones’ machine [F. B. Jones, Lect. Notes Math. 375, 149-152 (1974; Zbl 0286.54008)] they provide examples to show that neither implication can be reversed. The paper closes with two questions; one asks for an internal characterization of potential compactness, the other asks for a regular space without a dense completely regular subspace.
Reviewer: K.P.Hart (Delft)

MSC:

54D30 Compactness
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Citations:

Zbl 0286.54008
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