# Article

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Keywords:
\$1\$-paracompactness of \$Y\$ in \$X\$; \$2\$-paracompactness of \$Y\$ in \$X\$; \$1\$-collectionwise normality of \$Y\$ in \$X\$; \$2\$-collectionwise normality of \$Y\$ in \$X\$; \$1\$-normality of \$Y\$ in \$X\$; \$2\$-normality of \$Y\$ in \$X\$; quasi-\$P\$-embedding; quasi-\$C\$-embedding; quasi-\$C^{*}\$-embedding; \$1\$-metacompactness of \$Y\$ in \$X\$; \$1\$-subparacompactness of \$Y\$ in \$X\$
Summary:
Paracompactness (\$=2\$-paracompactness) and normality of a subspace \$Y\$ in a space \$X\$ defined by Arhangel'skii and Genedi [4] are fundamental in the study of relative topological properties ([2], [3]). These notions have been investigated by primary using of the notion of weak \$C\$- or weak \$P\$-embeddings, which are extension properties of functions defined in [2] or [18]. In fact, Bella and Yaschenko [8] characterized Tychonoff spaces which are normal in every larger Tychonoff space, and this result is essentially implied by their previous result in [8] on a corresponding case of weak \$C\$-embeddings. 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 Bella and Yaschenko above, 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\$. This is a negative answer to a question of Qu and Yasui in [25].
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