On applying uniform structures to study of free topological algebras
By a theorem of \textit{E. C. Nummela} [Topology Appl. 13, 77-83 (1982; Zbl 0471.22001)] and \textit{V. G. Pestov} [Vestn. Mosk. Univ., Ser. I 1982, No. 1, 35-37 (1982; Zbl 0499.22001)], for every pseudocompact space \(X\), a free topological group \(F(X)\) embeds canonically into the free topological group \(F(\beta X)\) over the Čech-Stone compactification \(\beta X\) of \(X\). By a theorem of the reviewer [Sib. Math. J. 28, No. 5, 824-832 (1987); translation from Sib. Mat. Zh. 28, No. 5(165), 167-177 (1987; Zbl 0631.22004)], if \(X^ n\) is countably compact and normal for each integer \(n\), then the free group \(F(X)\) is the inductive limit of its closed subspaces \(F(X)_ k\), consisting of words of reduced length \(\leq k\). In other words, a subset \(O\) of \(F(X)\) is open iff the intersection \(O\cap F(X)_ k\) is open in \(F(X)_ k\) for each \(k\geq 0\). The authors show that analogous results are valid for certain free algebras over Tikhonov spaces.
scientific article; zbMATH DE number 130863
scientific article