Free topological groups over metrizable spaces
The structure of the free topological group F(X) over a metrizable space X as well as the structure of the Abelian free topological group A(X) over X are investigated. The main results obtained by the authors can be summarized as follows: (1) the tightness of F(X) is countable iff X is separable or discrete; (2) F(X) is a k-space iff X is locally compact or discrete; (3) the tightness of A(X) is countable iff the set \(X'\) (of all non-isolated points of X) is separable; (4) A(X) is a k-space iff X is locally compact and \(X'\) is separable. This yields the negative answer to a question raised by \textit{M. M. Choban} [Tr. Mosk. Mat. O.-va. 48, 106- 149 (1985; Zbl 0606.22003)].
scientific article; zbMATH DE number 4129596
scientific article