Remarks on spaces with special type of \(k\)-networks
type: Item
instance of: scholarly article
title: Remarks on spaces with special type of \(k\)-networks
zbMATH Open document ID: 0886.54023
author: Masami Sakai
published in: Tsukuba Journal of Mathematics
publication date: 1998-05-04T00:00:00Z
review text: A cover \({\mathcal P}\) of a space is a \(k\)-network if, for any compact subset \(K\) and any open subset \(U\) with \(K\subset U\), there exists a finite \({\mathcal P}^* \subset {\mathcal C}\) such that \(K\subset \bigcup {\mathcal P}^* \subset U\). This paper gives negative answers to the following questions posed by \textit{Y. Ikeda} and \textit{Y. Tanaka} [Topol. Proc. 18, 107-132 (1993; Zbl 0829.54018)] by means of subspaces of the Stone-Čech compactifications of discrete spaces. These questions have positive answers if \(X\) is a \(k\)-space, and so on. Spaces are completely regular and \(T_1\). (a) Does every closed image of a space \(X\) with a star-countable \(k\)-network have a (star-countable \(k\)-network, or) point-countable \(k\)-network? (b) Is every space \(X\) with a locally countable \(k\)-network a \((\sigma\)-space, or) space in which every closed subset is a \(G_\delta\)-set?
Mathematics Subject Classification ID: 54D50
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 1102900
zbMATH Keywords: point-countable cover
zbMATH Keywords: \(k\)-network
zbMATH Keywords: \(k\)-space
reviewed by: Y. Tanaka
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163251
OpenAlex ID: W1568755176
DOI: 10.21099/TKBJM/1496163251
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description: scientific article; zbMATH DE number 1102900