Lower continuous topological groups

Abstract

A Hausdorff topological group G is called lower continuous if the topology of G has no predecessor in G2(G). The class of lower continuous topological groups contains all closed subgroups of products of minimal abelian groups, so strictly extend the class of minimal groups. Our main concern in this paper is the study of properties of lower continuous topological groups. Similar with the case for minimal groups, we provide a lower continuity criterion: a dense subgroup H of a Hausdorff topological abelian group G is lower continuous if and only if G is lower continuous and Soc(G)≤H. It is shown that every totally lower continuous abelian group is precompact. It is also shown that for a compact abelian groups G, G is hereditarily lower continuous if and only if G is torsion-free.