Variations on \(\omega\)-boundedness
type: Item
instance of: scholarly article
title: Variations on \(\omega\)-boundedness
zbMATH Open document ID: 1272.54024
author: István Juhász
author: Jan van Mill
author: William A. R. Weiss
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: Let \({\mathcal P}\) be a class of topological spaces. The authors call a Tychonoff space \(X\) \({\mathcal P}\)-bounded if every subspace of \(X\) belonging to \({\mathcal P}\) has a compact closure. For \({\mathcal P}=\) \{countable spaces\}, \({\mathcal P}\)-boundedness coincides with \(\omega\)-boundedness as introduced by \textit{S. L. Gulden}, \textit{W. F. Fleischman} and \textit{J. H. Weston} [Proc. Am. Math. Sec. 24, 197--203 (1970; Zbl 0203.55104)]. For \({\mathcal P}=\) \{\(\sigma\)-compact spaces\}, \({\mathcal P}\)-boundedness is identical with strong \(\omega\)-boundedness as introduced by P. Nyikos. The authors prefer to call these spaces \(\sigma\)C-bounded. Similarly, if \({\mathcal P}=\) \{weakly Lindelöf-spaces\} (respectively, = \{Lindelöf-spaces\}), then the \({\mathcal P}\)-bounded spaces are called wL-bounded (respectively, L-bounded). Obviously, every wL-bounded space is L-bounded, every L-bounded space is \(\sigma\)C-bounded, and every \(\sigma\)C-bounded space is \(\omega\)-bounded. In response to a question of \textit{P. Nyikos}, \textit{L. F. Aurichi} has constructed a locally compact and first countable \(\omega\)-bounded space that is not \(\sigma\)C-bounded in [Topology Appl. 156, No. 4, 775--782 (2009, Zbl 1162.54017)]. In this paper, the authors provide an example of a space that is \(\sigma\)C-bounded but not L-bounded. Whether there exists an L-bounded space that is not wL-bounded remains an open question. However, it is shown that, for locally compact spaces, wL-boundedness, L-boundedness, and \(\sigma\)C-boundedness are equivalent. For \({\mathcal P}=\) \{spaces satisfying the countable chain condition\}, \({\mathcal P}\)-bounded spaces are called ccc-bounded, and for \({\mathcal P}=\) \{hereditarily Lindelöf spaces\}, \({\mathcal P}\)-bounded spaces are called HL-bounded. Obviously, every wL-bounded space is ccc-bounded, every ccc-bounded space is HL-bounded, and every HL-bounded space is \(\omega\)-bounded. It is shown that the…
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54D45
zbMATH DE Number: 6176740
zbMATH Keywords: \(\omega\)-bounded space
zbMATH Keywords: strongly \(\omega\)-bounded space
zbMATH Keywords: wL-bounded space
zbMATH Keywords: L-bounded space
zbMATH Keywords: HL-bounded space
zbMATH Keywords: ccc-bounded space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: s11856-012-0062-8
OpenAlex ID: W2085036514
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DOI: 10.1007/S11856-012-0062-8
description: scientific article; zbMATH DE number 6176740