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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)

##### MSC:
 54C25 Embedding 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
##### Keywords:
relative topological properties; almost compact
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