1
$\begingroup$

A topological group is called minimal if it admits no strictly weaker Hausdorff group topology. By Prodanov-Stoyanov Theorem, a complete Abelian topological group is minimal if and only if it is compact. Since each non-compact complete topological group contains a non-compact closed separable subgroup, we obtain the following

Fact: A complete Abelian topological group $X$ is minimal if and only if each closed separable subgroup of $X$ is minimal.

Therefore, the minimality of complete Abelian toplogical groups is recognizable at the level of closed separable subgroups. Is the same true for non-commutaive groups?

Problem. Is a complete topological group $X$ minimal if each closed separable (or $\omega$-narrow) subgroup of $X$ is minimal?

A topological group $X$ is called $\omega$-narrow if for any neighborhood $U\subset X$ of the unit there exists a countable subset $A\subset X$ such that $X=AU=UA$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.