Continuity in topological groups
type: Item
instance of: scholarly article
title: Continuity in topological groups
zbMATH Open document ID: 1093.22001
published in: Topology and its Applications
publication date: 2006-04-06T00:00:00Z
review text: For a group \(G\) equipped with a topology \(\tau\), there is much work devoted to finding conditions on the group \(G\) yielding that \(G\) is a topological group. The authors consider groups \(G\) equipped with a Baire metrizable topology \(\tau\) and are interested in finding weaker conditions on the left and right translations that are still sufficient to characterize the group topology. In fact, they obtain the following result as main theorem. Theorem 1. Let \(G\) be a group which is a Baire metrizable space. Suppose that there is a dense subset \(S\) of second category in \(G\) such that the right translations \(\rho_s\) and \(\rho_{s^{-1}}\) are continuous for all \(x\in S\). Suppose further that, for each \(s\in G\), there is a residual subset \(R_s\) of \(G\) such that the left translation \(\lambda_x\) is almost-continuous on \(R_s\), \(\lambda_s(R_x)\) is residual, and \(\lambda_{s^{-1}}\) is almost-continuous on \(\lambda_s(R_x)\). Then \(G\) is a topological group. As an application, they show that if \(G\) is a second countable locally compact right topological group, then its topological center is a topological group.
reviewed by: Kohzo Yamada
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 5017445
zbMATH Keywords: topological group
zbMATH Keywords: Baire space
zbMATH Keywords: almost-continuous mapping
zbMATH Keywords: continuity of group operations
author: Stefano Ferri
author: Salvador Hernández
author: T. S. Wu
MaRDI profile type: Publication
full work available at URL: j.topol.2005.04.007
OpenAlex ID: W1973331183
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DOI: 10.1016/J.TOPOL.2005.04.007
description: scientific article; zbMATH DE number 5017445