On the extent of star countable spaces
type: Item
instance of: scholarly article
title: On the extent of star countable spaces
zbMATH Open document ID: 1246.54017
author: Lucia R. Junqueira
author: Jan van Mill
author: Richard G. Wilson
author: Ofelia T. Alas
author: Vladimir Tkachuk
published in: Central European Journal of Mathematics
publication date: 2012-01-16T00:00:00Z
review text: For a topological property \(\mathcal P\), a space \(X\) is \textit{star \(\mathcal P\)} if for every open cover \(\mathcal U\) of \(X\) there exists \(Y \subset X\) such that \(X=\text{St}(Y,\mathcal U) = \bigcup\{U\in\mathcal U: U\cap Y\neq\emptyset\}\), and \(Y\) has property \(\mathcal P\). Continuing the research done by \textit{O. T. Alas, L. R. Junqueira} and \textit{R. G. Wilson} in [Topology Appl. 158, No. 4, 620--626 (2011; Zbl 1226.54023)], the authors study star countable and star Lindelöf spaces establishing, among other things, that there exist first countable pseudocompact spaces which are not star Lindelöf. Also, some classes of spaces are described where star countability is equivalent to countable extent, and it is shown that a star countable space with a dense \(\sigma \)-compact subspace can have arbitrary extent. It is proved that for any \(\omega _{1}\)-monolithic compact space \(X\), if \(C _{p }(X)\) is star countable, then it is Lindelöf. The paper concludes with a list of related open problems.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54D60
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 5997913
zbMATH Keywords: Lindelöf property
zbMATH Keywords: extent
zbMATH Keywords: star properties
zbMATH Keywords: star countable spaces
zbMATH Keywords: star Lindelöf spaces
zbMATH Keywords: pseudocompact spaces
zbMATH Keywords: countably compact spaces
zbMATH Keywords: function spaces
zbMATH Keywords: \(\kappa\)-monolithic spaces
zbMATH Keywords: products of ordinals
zbMATH Keywords: \(\mathcal P\)-spaces
zbMATH Keywords: metalindelöf spaces
zbMATH Keywords: discrete subspaces
zbMATH Keywords: open expansions
reviewed by: Laszlo Zsilinszky
MaRDI profile type: Publication
full work available at URL: s11533-011-0018-y
OpenAlex ID: W2115255907
cites work: Countability and star covering properties
cites work: Q3049484
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cites work: Star covering properties
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cites work: Classes defined by stars and neighbourhood assignments
cites work: Q3732534
cites work: Monolithic spaces and \(D\)-spaces revisited
cites work: Combinatorial set theory
DOI: 10.2478/S11533-011-0018-Y
description: scientific article; zbMATH DE number 5997913