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$1$-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 $X$
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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