Three-space properties in paratopological groups
type: Item
instance of: scholarly article
title: Three-space properties in paratopological groups
zbMATH Open document ID: 1356.54036
author: D. Kharzeev
published in: Topology and its Applications
publication date: 2016-12-22T00:00:00Z
review text: A (semitopological) paratopological group is a group \(G\) with a topology such that just the the multiplication in \(G\) is (separately) continuous and the inversion in \(G\) is not necessarily continuous. A topological-algebraic property \(P\) is a three-space property in the class of semitopological (paratopological) groups provided that for every semitopological (paratopological) group \(G\) and a closed invariant subgroup \(N\) of \(G\), the fact that both \(N\) and \(G/N\) have \(P\) implies that \(G\) also has \(P\). In the paper the authors show that neither first-countable nor second-countable are three-space properties in the class of paratopological groups. They present a countable regular paratopological abelian group \(H\) which contains a closed discrete subgroup \(F\) such that \(H/F\) is topologically isomorphic to the rational numbers with the Sorgenfrey topology and \(H\) is not first-countable and prove that there exists a countable regular paratopological abelian group \(H\) which contains a closed discrete subgroup \(F\) such that \(H/F\) is topologically isomorphic to the rational numbers with the Sorgenfrey topology. However, \(H\) is not first-countable.
reviewed by: Osman Mucuk
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 6668173
zbMATH Keywords: three-space property
zbMATH Keywords: paratopological group
zbMATH Keywords: first-countable
zbMATH Keywords: second-countable
zbMATH Keywords: countable network
MaRDI profile type: Publication
full work available at URL: j.topol.2016.11.011
OpenAlex ID: W2554804730
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DOI: 10.1016/J.TOPOL.2016.11.011
description: scientific article; zbMATH DE number 6668173