COMPACT-LIKE TOTALLY DENSE SUBGROUPS OF COMPACT-GROUPS

A subgroup H of a topological group G is (weakly) totally dense in G if for each closed (normal) subgroup N of G the set H the-intersection-of N is dense in N. We show that no compact (or more generally, omega-bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group G is compact if and only if each continuous homomorphism pi: G --> H of G onto a topological group H is open. Here "Abelian" cannot be dropped. A connected, compact group contains a proper, weakly totally dense, countably compact subgroup if and only if its center is not a G(delta)-subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal G(delta)-subgroups is torsion. Under Lusin's hypothesis 2(omega)1 = 2(omega) the converse is true for a compact Abelian group G. If G is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup K of G and a proper, totally dense subgroup H of G with K is-contained-in-or-equal-to H (in particular, H is pseudocompact).