A theorem on remainders of topological groups
type: Item
instance of: scholarly article
title: A theorem on remainders of topological groups
zbMATH Open document ID: 1362.54031
author: Jan van Mill
author: A. V. Arhangel'skii
published in: Topology and its Applications
publication date: 2017-03-16T00:00:00Z
review text: Recall that a \textit{remainder} of a Tychonoff topological space \(X\) is the subspace \(bX\setminus X\) of a Hausdorff compactification \(bX\) of \(X\). Moreover, \(X\) is \textit{perfect} if every closed subset of \(X\) is a \(G_\delta\)-set of \(X\). By a result in [\textit{A. V. Arhangel'skii}, Fundam. Math. 203, No. 2, 165--178 (2009; Zbl 1182.54043)], for a remainder \(Y\) of a topological group \(G\), if \(Y\) is perfect then \(Y\) is first-countable. The main theorem of the present paper extends this result in case \(G\) is non-locally compact, by showing that \(Y\) perfect implies that \(G\) is separable and metrizable, \(Y\) is hereditarily Lindelöf and first-countable; moreover, \(Y\) is a \(p\)-space with countable \(\pi\)-base and every compact subspace of \(Y\) has a countable base of open neighborhoods in \(Y\). An example shows that \(Y\) needs not be necessarily metrizable. Moreover, it is known that there exists a non-locally compact topological group that is not metrizable but admits a first-countable remainder. Consequences of the main theorem are that a remainder of a topological group is perfect precisely when it is hereditarily Lindelöf (this answers Problem 3.9 from [loc. cit.]), and that a non-locally compact topological group is separable and metrizable if and only if it admits a perfect remainder.
reviewed by: Anna Giordano Bruno
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 6695391
zbMATH Keywords: remainder
zbMATH Keywords: compactification
zbMATH Keywords: topological group
zbMATH Keywords: first-countable
zbMATH Keywords: perfect
zbMATH Keywords: metrizable
zbMATH Keywords: hereditarily Lindelöf
MaRDI profile type: Publication
full work available at URL: j.topol.2017.02.003
OpenAlex ID: W2587670807
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DOI: 10.1016/J.TOPOL.2017.02.003
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description: scientific article; zbMATH DE number 6695391