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Some relative properties on normality and paracompactness, and their absolute embeddings. (English) Zbl 1121.54018
Summary: In this paper, we introduce notions of $$1$$-normality and $$1$$-collectionwise normality of a subspace $$Y$$ in a space $$X$$, which are closely related to $$1$$-paracompactness of $$Y$$ in $$X$$. Furthermore, notions of quasi-$$C^\ast$$- and quasi-$$P$$-embeddings are newly defined. Concerning the result of A. Bella and I. V. Yaschenko, by characterizing absolute cases of quasi-$$C^*$$- and quasi-$$P$$-embeddings, we obtain the following result: a Tychonoff space $$Y$$ is $$1$$-normal (or equivalently, $$1$$-collectionwise normal) in every larger Tychonoff space if and only if $$Y$$ is normal and almost compact. As another concern, we also prove that a Tychonoff (respectively, regular, Hausdorff) space $$Y$$ is $$1$$-metacompact in every larger Tychonoff (respectively, regular, Hausdorff) space if and only if $$Y$$ is compact. Finally, we construct a Tychonoff space $$X$$ and a subspace $$Y$$ such that $$Y$$ is $$1$$-paracompact in $$X$$ but not $$1$$-subparacompact in $$X$$.

##### MSC:
 54B10 Product spaces in general topology 54B05 Subspaces in general topology 54C20 Extension of maps 54C45 $$C$$- and $$C^*$$-embedding
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