  A topological group G is h-complete if every continuous homomorphic image of
G is (Raikov-)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 subdirectly
represented groups. Numerous applications are presented, among them: 1. Every
hereditarily h-complete group with quasi-invariant basis is the projective
limit of its metrizable 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.
