Set-Menger and related properties
type: Item
instance of: scholarly article
title: Set-Menger and related properties
zbMATH Open document ID: 1437.54020
author: Şükran Konca
author: Ljubiša D. R. Kočinac
published in: Topology and its Applications
publication date: 2020-04-07T00:00:00Z
review text: Let \(\mathcal P\) be a collection of subsets of a space \(X\) and say that \(X\) is \(\mathcal P\)-Menger (resp. weakly \(\mathcal P\)-Menger, almost \(\mathcal P\)-Menger) if for each \(A\in\mathcal P\) and each sequence \(\langle\mathcal U_n\rangle\) of families of open subsets such that \(\overline{A}\subset\cup\mathcal U_n\) for each \(n\) there is a sequence \(\langle\mathcal V_n\rangle\) such that each \(\mathcal V_n\) is a finite subset of \(\mathcal U_n\) and \(A\subset\cup_{n\in\omega}\cup\mathcal V_n\) (resp. \(A\subset\overline{\cup_{n\in\omega}\cup\mathcal V_n}\), \(A\subset\cup_{n\in\omega}\overline{\cup\mathcal V_n}\)); if each \(\mathcal V_n\) can be taken as a singleton then the word ``Menger'' is replaced by ``Rothberger.'' When \(\mathcal P=\{X\}\) then \(\mathcal P\) may be dropped and when \(\mathcal P\) is the power set of \(X\) then \(X\) is said to be set-Menger, etc. Hurewicz and Gerlits-Nagy versions are also given but the emphasis is on set-Menger and set-Rothberger, especially their behaviour under subspaces and (pre-)images. Topological games are described and it is shown that if \(X\) is Lindelöf then \(X\) is (weakly, almost) set-Menger if and only if the first player does not have a winning strategy in the respective game.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54G10
Mathematics Subject Classification ID: 91A44
zbMATH DE Number: 7187643
zbMATH Keywords: selection principles
zbMATH Keywords: game theory
zbMATH Keywords: Menger
zbMATH Keywords: set-Menger
zbMATH Keywords: quasi-Menger
zbMATH Keywords: P-space
reviewed by: David B. Gauld
MaRDI profile type: Publication
full work available at URL: j.topol.2019.106996
OpenAlex ID: W2996085014
cites work: Q5532970
cites work: Q4852557
cites work: Weak covering properties and infinite games
cites work: Weak covering properties and selection principles
cites work: Pixley-Roy spaces over subsets of the reals
cites work: A note on quasi-Menger and similar spaces
cites work: Q3994935
cites work: Q3652130
cites work: Q4804305
cites work: Q3935831
cites work: Weaker forms of the Menger property
cites work: Some weak covering properties and infinite games
cites work: The combinatorics of open covers
cites work: Ultrafilter Invariants in Topological Spaces
cites work: Q5401087
cites work: Q5613067
Wikidata QID: Q126584247
DOI: 10.1016/J.TOPOL.2019.106996
description: scientific article; zbMATH DE number 7187643