Topological groups with thin generating sets

A discrete subset S of a topological group G with identity 1 is oiled suita ble for G if S generates a dense subgroup of G and S boolean OR{1} is close d 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 Y-c,Y- Y-g and Y-cg of topological groups having a closed, generating and a closed generating sui table set are considered. The problem of stability of these classes under t he product, direct sum operations and taking subgroups or quotients is inve stigated. We show that (totally) minimal Abelian groups often have a suitab le set. It is also proved that every Abelian group endowed with the finest totally bounded group topology has a closed generating suitable set. More g enerally, the Bohr topology of every locally compact Abelian group admits a suitable set. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 22A 05; 54H11; secondary 22D05; 54A253 54D65.