Free topological groups of spaces and their subspaces
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 \(\text{ind} F_M(X)=0\).
scientific article; zbMATH DE number 1395038
scientific article