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Relative normality and product spaces. (English) Zbl 1097.54013
Summary: A. V. Arhangel’skii [Topology Appl. 70, 87–99 (1996; Zbl 0848.54016)], as one of various notions on relative topological properties, defined strong normality of $$A$$ in $$X$$ for a subspace $$A$$ of a topological space $$X$$, and showed 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 $$X$$, a 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.
##### MSC:
 54B10 Product spaces in general topology 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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