Automatic continuity of group operations
type: Item
instance of: scholarly article
title: Automatic continuity of group operations
zbMATH Open document ID: 0882.22001
DOI: 10.1016/S0166-8641(96)00119-8
published in: Topology and its Applications
publication date: 1998-03-02T00:00:00Z
review text: A subset \(A\) of a topological space \(X\) is said to have the Baire property if there is an open set \(U\subseteq X\) with \(A\Delta U\) meagre. Call a function \(f: C\to Y\) Baire measurable if, for any open set \(U\subseteq Y\), \(f^{-1}(U)\) has the Baire property. It is proved that if \((G,\cdot)\) is a group with a metric, separable and Baire topology \(\tau\) such that \(h\to g\cdot g\) is continuous for all \(g\in G\) and \(g\to g\cdot h\) is Baire measurable for all \(h\in G\), then \((G,\tau)\) is a topological group. As a consequence of this result, the author also considers the standard Borel groups. Some results of \textit{A. S. Kechris} [Classical descriptive set theory. Graduate Texts in Math. 156 (Berlin 1995; Zbl 0819.04002)] concerning Polish groups and free actions of standard Borel groups are established.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 57S99
zbMATH DE Number: 1022201
zbMATH Keywords: automatic continuity
zbMATH Keywords: semitopological group
zbMATH Keywords: Baire property
zbMATH Keywords: Baire measurable
zbMATH Keywords: standard Borel groups
zbMATH Keywords: Polish groups
author: Sławomir Solecki
author: Shashi Mohan Srivastava
reviewed by: K. P. Shum
MaRDI profile type: Publication
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Wikidata QID: Q126634805
full work available at URL: s0166-8641(96)00119-8
OpenAlex ID: W1969761980
description: scientific article; zbMATH DE number 1022201