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Paracompactness and closed subsets. (English) Zbl 0711.54017
J. Abdelhay [Gaz. Mat., Lisboa 9, No.37-38, 8-9 (1948)] observed that a Hausdorff space X is regular if and only if for every open cover \({\mathcal U}\) of X and every point x in X there exists a refinement \({\mathcal V}\) of \({\mathcal U}\) which is locally finite at x. He also gave an analogous characterization for normal spaces. The present paper gives some further characterizations along the same lines. Theorem 1.4 states: For X regular and E closed, E is compact if and only if every open cover \({\mathcal U}\) of X has an open refinement \({\mathcal V}\) such that \(\{\) \(V\in {\mathcal V}:\) \(V\cap E\neq \emptyset \}\) is finite. Theorem 2.2 states: For X Hausdorff, X is lightly compact \((=\) every locally finite family of open subsets of X is finite) iff every \(\alpha\)-paracompact \(E\subset X\) is compact (E is \(\alpha\)-paracompact in the sense of C. Aull [General Topology Relations modern Analysis Algebra 2, Proc. 2nd Prague topol. Sympos. 1966, 45-51 (1967; Zbl 0162.264)] provided every cover of E by sets open in X has a refinement by open subsets of X, which is locally finite in X, and which covers E).
Reviewer: J.E.Vaughan

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B05 Subspaces in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D30 Compactness
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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