The quasi-Rothberger property of linearly ordered spaces
type: Item
instance of: scholarly article
title: The quasi-Rothberger property of linearly ordered spaces
zbMATH Open document ID: 1404.54019
author: Zuquan Li
published in: Topology and its Applications
publication date: 2018-08-24T00:00:00Z
review text: Recall that a space \(X\) is said to be Rothberger if for each sequence \(\{\mathscr{U}_n: n\in \mathbb{N}\}\) of open covers of \(X\), there is a \(U_n\in \mathscr{U}_n\) for each \(n\in \mathbb{N}\) such that \(\{U_n: n\in \mathbb{N}\}\) covers \(X\); a space \(X\) is said to be quasi-Rothberger if for each closed set \(F\subset X\) and each sequence \(\{\mathscr{U}_n: n\in \mathbb{N}\}\) of open covers of \(F\) by sets open in \(X\), there is a \(U_n\in \mathscr{U}_n\) for each \(n\in \mathbb{N}\) such that \(F\subset \overline{\bigcup_{n\in \mathbb{N}}U_n}\); a space \(X\) is said to be quasi-Lindelöf if for each closed set \(F\subset X\) and each open cover \(\mathscr{U}\) of \(F\) by open sets of \(X\), there is a countable family \(\{U_n: n\in \mathbb{N}\}\) of \(\mathscr{U}\) such that \(F\subset \overline{\bigcup_{n\in \mathbb{N}}U_n}\). Clearly, the quasi-Rothberger property is weaker than the Rothberger property and stronger than the quasi-Lindelöf property; the Lindelöf property is weaker than the Rothberger property and stronger than the quasi-Lindelöf property. There are no implications between the quasi-Rothberger property and the Lindelöf property (see Example 1.2 and Example 1.3 of the paper). In the paper under review the author studies the quasi-Rothberger property and obtains mainly the following results. (1) For a metrizable space \(X\), the quasi-Rothberger property, the quasi-Lindelöf property and separability are equivalent. (2) Let \((X, <) \) be a linearly ordered space of cardinality \(\alpha\) with \(cf(\alpha)>\omega\); then the following are equivalent: (a) \(X\) is quasi-Rothberger; (b) \(X\) is quasi-Lindelöf; (c) \(e(X)=\omega\) and each closed well-ordered (resp., closed anti-well-ordered) subset of \(X\) of order type \(\alpha\) with \(cf(\alpha)>\omega\) has a largest (resp., a smallest) element, where for a well-ordered set \((X,<)\), the converse order \(>\) of \(<\) is said to be anti-well-ordered of \((x,<)\). The author poses…
reviewed by: Yin-Zhu Gao
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 6924612
zbMATH Keywords: quasi-Lindelöf
zbMATH Keywords: quasi-Rothberger
zbMATH Keywords: extent
zbMATH Keywords: anti-well-ordered
zbMATH Keywords: linearly ordered space
MaRDI profile type: Publication
full work available at URL: j.topol.2018.07.011
OpenAlex ID: W2883721857
cites work: Q4852557
cites work: Diagonalizations of dense families
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cites work: A note on quasi-Menger and similar spaces
cites work: Q3994935
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Wikidata QID: Q129488321
DOI: 10.1016/J.TOPOL.2018.07.011
description: scientific article; zbMATH DE number 6924612