A note on the Menger (Rothberger) property and topological games
type: Item
instance of: scholarly article
title: A note on the Menger (Rothberger) property and topological games
zbMATH Open document ID: 1333.54034
author: Liang-Xue Peng
author: Zhan-Chao Shen
published in: Topology and its Applications
publication date: 2016-01-21T00:00:00Z
review text: Making use of several topological games, a number of conditions either equivalent to or implying either the Menger or Rothberger property are presented. As an example it is shown that a \(T_1\) space \(X\) is Menger (Rothberger) if and only if for each sequence \(\langle\phi_n\rangle\) of neighbourhood assignments for \(X\) and each \(n\) there is a finite (singleton) set \(D_n\subset X\) so that \(X=\bigcup\{\phi_n(d) \mid d\in D_n, n\in\omega\}\) and \(\bigcup_{n\in\omega} D_n\) is discrete and closed in \(X\). \(\mathbf K\)-like spaces are also considered for certain classes \(\mathbf K\). For example if \(\mathbf K\) is the class of all spaces with the Menger (Rothberger) property then a \(\mathbf K\)-like space is Menger (Rothberger).
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 91A44
zbMATH DE Number: 6534253
zbMATH Keywords: Menger space
zbMATH Keywords: Rothberger space
zbMATH Keywords: \(\mathbf K\)-like
zbMATH Keywords: neighbourhood assignment
zbMATH Keywords: selection properties, topological games
reviewed by: David B. Gauld
MaRDI profile type: Publication
full work available at URL: j.topol.2015.12.059
OpenAlex ID: W2219104656
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DOI: 10.1016/J.TOPOL.2015.12.059
description: scientific article; zbMATH DE number 6534253