Paracompactness and closed subsets.

*(English)*Zbl 0711.54017J. 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.) |