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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...

Arhangel’skiĭ 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...