Equalities for the extent of infinite products and \(\varSigma\)-products
type: Item
instance of: scholarly article
title: Equalities for the extent of infinite products and \(\varSigma\)-products
zbMATH Open document ID: 1487.54013
author: Yasushi Hirata
author: Toshimichi Usuba
author: Yukinobu Yajima
published in: Topology and its Applications
publication date: 2022-01-26T00:00:00Z
review text: For a topological cardinal function \(\varphi\), it is an interesting problem to ask when \(\varphi(X)= |\Lambda| \cdot \sup \{ \, \varphi (X_ \lambda ) \mid \lambda \in \Lambda \,\}\), for some product \(X = \prod _{ \lambda \in \Lambda} X_ \lambda\) of topological spaces [\textit{I. Juhász,} Cardinal functions in topology -- ten years later. Amsterdam: Mathematisch Centrum (1980; Zbl 0479.54001)]. Motivated by a comparison with known results about tightness, the authors study the above problem with respect to extent, as well as a similar problem concerning \(\Sigma\)-products. In detail, recall that the \emph{extent} of a topological space \(X\) is \[e(X) = \omega \cdot \sup \{ \, |D| \mid D \text{ is a closed discrete subset in } X \, \}. \] Compact factors in a product make no contribution to the extent of the product, hence all factors are supposed to be noncompact. All topological spaces are assumed to be regular and \(T_1\). For \(x \in X\) and a cover \(\mathcal U\) of \(X\), \( St(x, \mathcal U ) \) is \( \bigcup \{ \,U \in \mathcal U \mid x \in U \, \}\). A space \(X\) is a \emph{strict p-space} if there is a sequence \(\{ \mathcal U_n \}\) of open covers of \(X\) such that, for each \(x \in X\), both (i) \(\bigcap _{n \in \omega } St(x, \mathcal U _n)\) is compact, and (ii) for each open set \(U\) in \(X\) such that \(\bigcap _{n \in \omega } St(x, \mathcal U _n)\subset U\), there is an \( m \in \omega\) such that \( St(x, \mathcal U _m) \subset U\). A space \(X\) is a \emph{strong \(\Sigma\)-space} (resp., \emph{\(\Sigma\)-space}) if there are a sequence \(\{ \mathcal F_n \}\) of locally finite closed covers of \(X\) and a cover \(\mathcal K\) of \(X\) by compact (resp., countably compact closed) sets such that for each \(K \in \mathcal K\) and each open set \(U \in X\) with \(K \subset U\), there is an \(F \in \bigcup _{n \in \omega } \mathcal F_n\) with \(K \subset F \subset U\). Among many other interesting results, the following theorems are proved. …
reviewed by: Paolo Lipparini
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54E18
zbMATH DE Number: 7464226
zbMATH Keywords: extent
zbMATH Keywords: product
zbMATH Keywords: \(\varSigma\)-product
zbMATH Keywords: strict \(p\)-space
zbMATH Keywords: strong \(\varSigma\)-space
zbMATH Keywords: semi-stratifiable space
zbMATH Keywords: strong \(\beta\)-space
zbMATH Keywords: \(p\)-space
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107946
OpenAlex ID: W4200601259
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DOI: 10.1016/J.TOPOL.2021.107946
description: scientific article; zbMATH DE number 7464226