ar X iv : m at h / 04 02 23 6 v 2 [ m at h . G N ] 4 A ug 2 00 4 Hereditarily h - complete groups ∗

A topological group G is h-complete if every continuous homomorphic image of G is (Ra˘ ıkov-)complete; we say that G is hereditarily h-complete if every closed subgroup of G is h-complete. In this paper, we establish open-map properties of hereditarily h-complete groups with respect to large classes of groups, and prove a theorem on the (total) minimality of sub-directly represented groups. Numerous applications are presented, among them: 1. Every hereditarily h-complete group with quasi-invariant basis is the projective limit of its metriz-able quotients; 2. If every countable discrete hereditarily h-complete group is finite, then every locally compact hereditarily h-complete group that has small invariant neighborhoods is compact. In the sequel, several open problems are formulated.