On \(s\)-images of metric spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 0962.54025
author: Huaipeng Chen
publication date: 2001-03-28T00:00:00Z
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54H05
zbMATH DE Number: 1566997
zbMATH Keywords: point-countable family
zbMATH Keywords: \(s\)-mapping
zbMATH Keywords: \(k\)-network
zbMATH Keywords: locally compact
MaRDI profile type: Publication
title: On \(s\)-images of metric spaces
published in: Topology Proceedings
review text: Let \(X\) be a space, and \({\mathcal P}\) a cover of \(X\). \({\mathcal P}\) is said to be a \(k\)-network for \(X\) if \(K\subset U\) with \(K\) compact and \(U\) open, then \(K\subset\bigcup {\mathcal P}'\subset U\) for some finite subset \({\mathcal P}'\) of \({\mathcal P}\). If \({\mathcal P}\) is a \(k\)-network for \(X\), then \({\mathcal P}\) is a closed (countably compact, compact) \(k\)-network if each element of \({\mathcal P}\) is closed (countably compact, compact) in \(X\). Every quotient \(s\)-image of a metric space, or every closed image of a metric space has a point-countable \(k\)-network. \textit{S. Lin} [Topol. Proc. 20, 185-190 (1995; Zbl 0869.54025)] proved that for a regular and \(T_1\)-space \(X\), if \(X\) has a point-countable closed \(k\)-network, then \(X\) has a point-countable countably compact \(k\)-network if and only if every first countable closed subset of \(X\) is locally compact. The following question was posed: Suppose a space \(X\) has a point-countable closed \(k\)-network. Is \(X\) a space with a point-countable compact \(k\)-network if every first countable closed subspace of \(X\) is locally compact?NEWLINENEWLINENEWLINEIn the paper under review the author discusses some relations of the \(s\)-images of metric spaces and the spaces with a point-countable closed \(k\)-network, and constructs an example which answers negatively to the above question by showing that there is a regular \(T_1\) countably compact space \(Y\) such that \(Y\) has a point-countable closed \(k\)-network and every first countable closed subspace of \(Y\) is compact, but \(Y\) has no point-countable compact \(k\)-network.
reviewed by: Shou Lin
description: scientific article