Local connectedness in topological groups
type: Item
instance of: scholarly article
title: Local connectedness in topological groups
zbMATH Open document ID: 1311.54027
author: Keith Whittington
published in: Topology and its Applications
publication date: 2015-01-06T00:00:00Z
review text: Many results about the conditions for locally compact groups to be locally path-connected were obtained and the interest of the study has turned to more general topological groups, for example, to Polish topological groups. In [Trans. Am. Math. Soc. 212, 393--400 (1975; Zbl 0318.54037)], \textit{G. S. Ungar} showed that every path-connected Polish topological group is locally path-connected. In [Proc. Am. Math. Soc. 130, No. 3, 903--907 (2002; Zbl 0985.54034)], the author showed that every topological group with a second countable, second category path component is locally connected. In this paper, the author gives basic principles underlying these results and, consequently, improves them. For instance, he proves the following results. 1. The following are equivalent for a topological group \(G\): (1) \(G\) is locally connected; (2) \(G\times G\) is locally connected; (3) \(G\) is diagonally connected, where a space \(X\) is \textit{diagonally connected} if for each open set \(V\) in \(X\times X\) containing the diagonal \(D\) of \(X\), there is a connected open set \(W\) such that \(D\subseteq W\subseteq V\); (4) \(G\) contains a subspace \(C\) that is diagonally connected and somewhere dense in \(G\). 2. If a topological group \(G\) is Polish, then the following are equivalent: (1) \(G\) is locally connected; (2) \(G\) is path-connected; (3) \(G\) has a path component that is second category in itself and somewhere dense in \(G\); (4) \(G\) is locally path-connected. Furthermore, if \(G\) is finite dimensional the following statements can be added: (5) \(G\) is locally compact and path-connected; (6) \(G\) is a Lie group. 3. If a topological group \(G\) has a second countable path component that is second category in itself and somewhere dense in \(G\), then \(G\) is almost locally path-connected, and hence locally connected.
reviewed by: Kohzo Yamada
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 22D45
zbMATH DE Number: 6383760
zbMATH Keywords: topological group
zbMATH Keywords: arc
zbMATH Keywords: path
zbMATH Keywords: locally connected
zbMATH Keywords: locally path connected
zbMATH Keywords: locally arc connected
zbMATH Keywords: locally compact
zbMATH Keywords: diagonal
zbMATH Keywords: diagonally connected
zbMATH Keywords: open mapping
zbMATH Keywords: almost open
zbMATH Keywords: Polish
zbMATH Keywords: transformation group
zbMATH Keywords: Lie group
zbMATH Keywords: Baire
MaRDI profile type: Publication
full work available at URL: j.topol.2014.11.006
OpenAlex ID: W1978286856
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DOI: 10.1016/J.TOPOL.2014.11.006
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description: scientific article; zbMATH DE number 6383760