A note on quasi-Menger and similar spaces
type: Item
instance of: scholarly article
title: A note on quasi-Menger and similar spaces
zbMATH Open document ID: 1303.54007
author: Giuseppe Di Maio
author: Ljubiša D. R. Kočinac
published in: Topology and its Applications
publication date: 2014-10-24T00:00:00Z
reviewed by: Samuel Gomes da Silva
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54E20
zbMATH DE Number: 6361147
zbMATH Keywords: selection principles
zbMATH Keywords: weakly Menger
zbMATH Keywords: almost Menger
zbMATH Keywords: quasi-Menger
zbMATH Keywords: Alster space
zbMATH Keywords: \(\gamma\)-\(k\)-space
MaRDI profile type: Publication
full work available at URL: j.topol.2014.08.024
OpenAlex ID: W2070620196
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DOI: 10.1016/J.TOPOL.2014.08.024
review text: Let \(\mathcal{A}\) and \(\mathcal{B}\) be families of sets, with \(\emptyset \notin \mathcal{A}\). The \textit{selection principles} \(S_1(\mathcal{A},\mathcal{B})\) and \(S_{\mathrm{fin}}(\mathcal{A},\mathcal{B})\) are stated as follows:NEWLINENEWLINE\(S_1(\mathcal{A},\mathcal{B})\) declares that for every sequence \(\langle \mathcal{A}_n: n < \omega \rangle\) of elements of \(\mathcal{A}\) there is a sequence \(\langle B_n: n < \omega\rangle\) such that \(B_n \in \mathcal{A}_n\) for all \(n < \omega\) and \(\{B_n: n < \omega\} \in \mathcal{B}\).NEWLINENEWLINE\(S_{\mathrm{fin}}(\mathcal{A},\mathcal{B})\) declares that for every sequence \(\langle \mathcal{A}_n: n < \omega \rangle\) of elements of \(\mathcal{A}\) there is a sequence \(\langle \mathcal{B}_n: n < \omega\rangle\) such that \(\mathcal{B}_n\) is a finite subset of \(\mathcal{A}_n\) for all \(n < \omega\) and \(\bigcup\{\mathcal{B}_n: n < \omega\} \in \mathcal{B}\).NEWLINENEWLINEThese selection principles are related to classical covering properties of Rothberger, Menger and Hurewicz. Let \(\mathcal{O}_X\) denote the family of all open covers of a topological space \(X\). \(X\) is said to be a \textit{Rothberger space} if \(S_1(\mathcal{O}_X,\mathcal{O}_X)\) holds, and it is said to be a \textit{Menger space} if \(S_{\mathrm{fin}}(\mathcal{O}_X,\mathcal{O}_X)\) holds.NEWLINENEWLINEIn the paper under review, the authors introduce a number of weaker forms of these classical covering properties.NEWLINENEWLINELet \(\mathcal{\overline{O}}\) be the family of all collections of open sets \(\mathcal{U}\) such that \(\{\overline{U}: U \in \mathcal{U}\}\) covers \(X\), and let \(\mathcal{O}_D\) be the family of all collections of open sets \(\mathcal{U}\) such that \(\bigcup \mathcal{U}\) is a dense subset of \(X\).NEWLINENEWLINE\(X\) is \textit{almost Rothberger (almost Menger)} if \(S_1(\mathcal{O}_X, \mathcal{\overline{O}})\) (respectively, \(S_{\mathrm{fin}}(\mathcal{O}_X, \mathcal{\overline{O}})\)) holds. \(X…
description: scientific article; zbMATH DE number 6361147