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On relatively normal spaces, relatively regular spaces, and on relative property \((a)\). (English) Zbl 0951.54017
The following results concerning relative topological properties are proved:
(1) A regular (Tychonoff) space \(Y\) is normal in every larger regular (Tychonoff) space if and only if \(Y\) is Lindelöf or normal almost compact.
(2) A functionally Hausdorff space \(Y\) is regular in every larger functionally Hausdorff space if and only if \(Y\) is compact.
(3) A Hausdorff (regular, Tychonoff) space \(Y\) is relatively (a) in every larger Hausdorff (regular, Tychonoff) space \(X\) if and only if \(Y\) is compact.
Reviewer: M.Ganster (Graz)

54C25 Embedding
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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