×

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.)

Citations:

Zbl 0848.54016
PDFBibTeX XMLCite
Full Text: EuDML EMIS