Normal covers of infinite products and normality of \(\sigma\) -products
type: Item
instance of: scholarly article
title: Normal covers of infinite products and normality of \(\sigma\) -products
zbMATH Open document ID: 1109.54010
author: Yukinobu Yajima
published in: Topology and its Applications
publication date: 2006-12-14T00:00:00Z
review text: This paper studies normal covers of a product of infinitely many \(\beta\)-spaces and the normality of a \(\Sigma\)-product of \(\beta\)-spaces. The notion of \(\beta\)-spaces was introduced by \textit{R. E. Hodel} [Pac. J. Math. 38, 641--652 (1971; Zbl 0219.54024)], and it is known that all \(\Sigma\)-spaces and all semi-stratifiable spaces are \(\beta\)-spaces. First, let \(X=\prod_{\lambda\in\Lambda}X_\lambda\) be an infinite product space, and define \(X_\theta=\prod_{\lambda\in\theta}X_\lambda\) for each \(\theta\subseteq\Lambda\). A subset \(A\) of \(X\) is called a \textit{cylinder} if there is a finite set \(\theta\subseteq\Lambda\) and \(T\subseteq X_\theta\) such that \(A=\pi_\theta^{-1}[T]\), where \(\pi_\theta:X\to X_\theta\) is the projection. A subset \(A\) of \(X\) is called a \textit{rectangle} if there is a finite set \(\theta\subseteq\Lambda\) and \(T_\lambda\subseteq X_\lambda\) \((\lambda\in\theta)\) such that \(A=\bigcap_{\lambda\in\theta}\pi_\lambda^{-1}[T_\lambda]\), where \(\pi_\lambda:X\to X_\lambda\) is the projection. The author proves a technical theorem which implies that (1) if each \(X_\lambda\) is a \(\beta\)-space and if \(X_\theta\) is paracompact and has countable tightness for every finite set \(\theta\subseteq\Lambda\), then an open cover \({\mathcal O}\) of \(X\) is normal if and only if \({\mathcal O}\) has a locally finite, \(\sigma\)-discrete, cylindrical cozero refinement which has a cylindrical zero-set shrinking. The following conditions (a)--(e) on a cover \({\mathcal O}\) of \(X\) are considered: (a) \({\mathcal O}\) is normal; (b) \({\mathcal O}\) has a \(\sigma\)-locally finite rectangular cozero refinement; (c) \({\mathcal O}\) has a \(\sigma\)-discrete rectangular cozero refinement; (d) \({\mathcal O}\) has a locally finite rectangular cozero refinement; (e) \({\mathcal O}\) has a locally finite, \(\sigma\)-discrete, rectangular cozero refinement which has a rectangular zero-set shrinking. As an application of (1), t…
reviewed by: Haruto Ohta
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E18
zbMATH DE Number: 5080157
zbMATH Keywords: product
zbMATH Keywords: normal cover
zbMATH Keywords: paracompact
zbMATH Keywords: \(\beta\)-space
zbMATH Keywords: \(\Sigma\)-space
zbMATH Keywords: \(M\)-space
MaRDI profile type: Publication
full work available at URL: j.topol.2006.02.004
OpenAlex ID: W2011136349
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DOI: 10.1016/J.TOPOL.2006.02.004
description: scientific article; zbMATH DE number 5080157