Topological groups with thin generating sets

Abstract

A discrete subset S of a topological group G with identity 1 is called suitable for G if S generates a dense subgroup of G and S ∪ {1} is closed in G. We study various algebraic and topological conditions on a group G which imply the existence of a suitable set for G as well as the restraints imposed by the existence of such a set. The classes S c , S g and S cg of topological groups having a closed, generating and a closed generating suitable set are considered. The problem of stability of these classes under the product, direct sum operations and taking subgroups or quotients is investigated. We show that (totally) minimal Abelian groups often have a suitable set. It is also proved that every Abelian group endowed with the finest totally bounded group topology has a closed generating suitable set. More generally, the Bohr topology of every locally compact Abelian group admits a suitable set.