Free topological groups of spaces and their subspaces
type: Item
instance of: scholarly article
title: Free topological groups of spaces and their subspaces
zbMATH Open document ID: 0943.22003
DOI: 10.1016/S0166-8641(98)00123-0
author: Ol'ga Sipacheva
published in: Topology and its Applications
publication date: 2000-08-21T00:00:00Z
review text: Free topological groups were introduced by \textit{A. A. Markov} [Dokl. Akad. Nauk SSSR 31, 299-301 (1941)]: for a Tychonoff topological space \(X\), \(F_M(X)\) is the algebraic free group of the set \(X\) equipped with the strongest group topology such that any continuous mapping of \(X\) to a topological group \(G\) can be extended to a continuous homomorphism \(F_M(X)\to G\). If \(Y\) is a subspace of \(X\) then the natural embedding \(F_M(Y)\to F_M(X)\) is continuous and algebraically isomorphic. The author gives a criterion for this embedding to be homeomorphic. Let \(X\) be a completely regular \(T_1\) space, \(Y\subseteq X\). The topology in the free group of the set \(Y\) induced by \(F_M(X)\) coincides with the topology of \(F_M(Y)\) if and only if each bounded continuous pseudometric on \(Y\) can be extended to a continuous pseudometric on \(X\). The sufficiency was proved by \textit{V. G.~Pestov} [Vestn. Mosk. Univ. (1) 1982, 35-37 (1982; Zbl 0499.22001)]. The author constructs a family of pseudometrics and uses a result of Markov to prove the necessity. As corollaries, the following results are derived: 1) if a completely regular \(T_1\) space \(X\) is Dieudonné complete then the group \(F_M(X)\) is Weil complete; 2) any \(T_0\) topological group \(G\) is a quotient group of a Weil complete \(T_0\) topological group; 3) if \(\dim X=0\) then \(\text{ind} F_M(X)=0\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 1395038
zbMATH Keywords: free topological group
zbMATH Keywords: pseudometric
reviewed by: Victor Gichev
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00123-0
OpenAlex ID: W2079533728
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description: scientific article; zbMATH DE number 1395038