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

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]

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

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]

Relative normality and product spaces

summary:Arhangel'ski\u{\i} defines in [Topology Appl. 70 (1996), 87--99], as one of various notions on relative topological properties, strong normality of $A$ in $X$ for a subspace $A$ of a topological space $X$, and shows that this is equivalent to normality of $X_A$, where $X_A$ denotes the space obtained from $X$ by making each point of $X \setminus A$ isolated. In this paper we investigate for a space $X$, its subspace $A$ and a space $Y$ the normality of the product $X_A \times Y$ in connection with the normality of $(X\times Y)_{(A\times Y)}$. The cases for paracompactness, more generally, for $\gamma$-paracompactness will also be discussed for $X_A\times Y$. As an application, we prove that for a metric space $X$ with $A \subset X$ and a countably paracompact normal space $Y$, $X_A \times Y$ is normal if and only if $X_A \times Y$ is countably paracompact

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

Dedicated to Professor Takao Hoshina on his 60th birthday. Abstract. 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 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 Classification: Primary 54B10; Secondary 54B05, 54C20, 54C45, 54D15, 54D2