Topologies on groups determined by discrete subsets
type: Item
instance of: scholarly article
title: Topologies on groups determined by discrete subsets
zbMATH Open document ID: 1149.54001
author: Yevhen Zelenyuk
published in: Topology and its Applications
publication date: 2008-06-24T00:00:00Z
review text: Let \(\mathcal F\) be a filter on an infinite group \(G\). Suppose that \(\mathcal F\) contains all subsets of \(G\) whose cardinality is strictly smaller than that of their complements, and there is a map \(M:G\to \mathcal F\) such that the subsets \(xM(x)\) are pairwise disjoint. Such filters are called strongly discrete by the author. Let \(\mathcal T[\mathcal F]\) be the largest left invariant topology on \(G\) for which \(\mathcal F\) converges to the identity \(e\). The author shows, among other results, that \((G,\mathcal T[\mathcal F])\) is zero-dimensional and Hausdorff and, if \(\mathcal F\) is in addition an ultrafilter, \((G,\mathcal T[\mathcal F])\) is extremally disconnected and has no local homeomorphisms other than local translations. Moreover, two strongly discrete filters \(\mathcal F\) and \(\mathcal G\) on \(G\) are isomorphic if and only if the topologies \(\mathcal T[\mathcal F]\) and \(\mathcal T[\mathcal G]\) are homeomorphic. Finally, the author shows the undecidability in ZFC of the question whether there exists on any countable group \(G\) a strongly discrete ultrafilter \(\mathcal F\) such that \(\mathcal T [\mathcal F]\) is a group topology.
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54G05
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 5292210
zbMATH Keywords: left invariant topology
zbMATH Keywords: extremally disconnected space
zbMATH Keywords: Ramsey ultrafilter
zbMATH Keywords: discrete subset
reviewed by: X. Domínguez
MaRDI profile type: Publication
full work available at URL: j.topol.2008.03.017
OpenAlex ID: W1997292586
cites work: Q5532970
cites work: Ultrafilters: Some old and some new results
cites work: A Topological Group Having no Homeomorphisms Other than Translations
cites work: Q3218775
cites work: Proper forcing
cites work: THE PRODUCT OF TOPOLOGICAL GROUPS AND EXTREMAL DISCONNECTEDNESS
cites work: On extremally disconnected topological groups
DOI: 10.1016/J.TOPOL.2008.03.017
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description: scientific article; zbMATH DE number 5292210