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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

##### MSC:
 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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