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.
@article{119405,
author = {Takao Hoshina and Ryoken Sokei},
title = {Relative normality and product spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {515-524},
zbl = {1097.54013},
mrnumber = {2025817},
language = {en},
url = {http://dml.mathdoc.fr/item/119405}
}
Hoshina, Takao; Sokei, Ryoken. Relative normality and product spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 515-524. http://gdmltest.u-ga.fr/item/119405/
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