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.