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On relative normality and relative symmetrizability. (English. Russian original) Zbl 0907.54012

Mosc. Univ. Math. Bull. 50, No. 3, 28-31 (1995); translation from Vestn. Mosk. Univ., Ser. I 1995, No. 3, 32-36 (1995).
This paper continues the study of relative separation properties and of relative covering properties. A subspace \(Y\) of a space \(X\) is said to be strongly pseudocompact in \(X\) if every family of (in \(X\)) open sets (where each set has nonempty intersection with \(Y\)), which is locally finite at each point of \(Y\), has to be finite. A subspace \(Y\) of \(X\) is called normal in \(X\), if for any disjoint closed subsets \(A\) and \(B\) of \(X\) there exist (in \(X\)) open sets \(U\) and \(V\) such that \(A\cap Y\subseteq U\) and \(B\cap Y\subseteq V\). The authors show that if a subspace \(Y\) is normal and strongly pseudocompact in \(X\) then it is countably compact in \(X\). Next the authors define the notion of a symmetric \(d\) on \(X\) defining a subspace \(Y\) of \(X\). They are able to generalize a result of S. Nedev in the following way: If a symmetric \(d\) on \(X\) defines a subspace \(Y\) and an extent of \(Y\) in \(X\) is countable then the subspace \(Y\) is perfect. Furthermore, they show that if \(X\) is symmetrizable and the subspace \(Y\) is strongly pseudocompact and normal in \(X\) then \(Y\) is compact and metrizable. Also, a result of R. M. Stephenson jun. is intensified by showing that, if \(X\) is semimetrizable and \(Y\) is countably compact in \(X\) then \(Y\) has a countable base in \(X\).
Reviewer: M.Ganster (Graz)

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54A35 Consistency and independence results in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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