Minimal and precompact group topologies on free groups
A topological group G is called minimal if its topology is a minimal element of the set of all Hausdorff group topologies on G. If G/N is minimal for every closed normal subgroup N of G, then G is called totally minimal. It was proved by \textit{D. Shakhmatov} [Mat. Zametki 38, 908-914 (1985); English transl.: Math. Notes 38, 1003-1006 (1985; Zbl 0594.22001)] that on every free group G of infinite rank there exists a totally minimal group topology \(\tau\). Since in this construction (G,\(\tau\)) has no group completion, it is natural to ask whether there exist totally minimal group topologies \(\tau\) such that (G,\(\tau\)) has a group completion. A more difficult problem is to construct a precompact totally minimal group topology \(\tau\). In the first part of the paper under review the author shows that on very countable free group there exist infinitely many nonisomorphic precompact totally minimal group topologies. The second part of the paper is devoted mainly to the existence of certain types of \(T_ 2\)-precompact group topologies on free groups and the study of the number of these topologies. It is proved, in particular, that every free group F with cardinality \(\alpha\) admits a \(T_ 2\)- precompact group topology \(\tau\) such that \(w(F,\tau)=\log \alpha\).
precompact totally minimal group topologies
scientific article; zbMATH DE number 4198341
scientific article