Some relative properties on normality and paracompactness, and their absolute embeddings.

*(English)*Zbl 1121.54018Summary: 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 \(X\).