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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.
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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