The normality of \(\Sigma\)-products and the perfect \(\kappa\)-normality of Cartesian products
type: Item
instance of: scholarly article
title: The normality of \(\Sigma\)-products and the perfect \(\kappa\)-normality of Cartesian products
zbMATH Open document ID: 0556.54007
author: Yukinobu Yajima
published in: Journal of the Mathematical Society of Japan
publication date: 1984-01-01T00:00:00Z
review text: The author first proves that, if \(\Sigma\) is a \(\Sigma\)-product of paracompact \(\Sigma\)-spaces, then \(\Sigma\) is collectionwise normal if and only if it is normal. This generalizes a theorem of \textit{A. P. Kombarov} [Sov. Math. Dokl. 19, 403-407; Dokl. Akad. Nauk SSSR 239, 775- 778 (1978; Zbl 0397.54013)]. Next it is proved that, if X is a Cartesian product of paracompact \(\sigma\)-spaces and if each finite subproduct of X has countable tightness, then X is perfectly \(\kappa\)-normal (that is, if for each disjoint open sets \(V_ 1\) and \(V_ 2\) in X there exist disjoint cozero-sets \(U_ 1\) and \(U_ 2\) in X such that \(V_ k\subset U_ k\), \(k=1,2)\). This generalizes a theorem of \textit{V. E. Shchepin} [Sov. Math. Dokl. 17, 152-155; Dokl. Akad. Nauk SSSR 226, 527-529 (1976; Zbl 0338.54022)]. A result about the countable paracompactness of \(\Sigma\)-products is also obtained.
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E18
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 3886718
zbMATH Keywords: collectionwise normality
zbMATH Keywords: perfectly \(\kappa \) -normal space
zbMATH Keywords: \(\Sigma \) - product
zbMATH Keywords: paracompact \(\Sigma \) -spaces
zbMATH Keywords: Cartesian product
zbMATH Keywords: paracompact \(\sigma \) -spaces
zbMATH Keywords: countable tightness
zbMATH Keywords: countable paracompactness of \(\Sigma \) -products
reviewed by: Tadashi Ishii
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03640689
OpenAlex ID: W2024017123
DOI: 10.2969/JMSJ/03640689
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description: scientific article; zbMATH DE number 3886718