Compact-like totally dense subgroups of compact groups

Abstract

A subgroup H H of a topological group G G is (weakly) totally dense in G G if for each closed (normal) subgroup N N of G G the set H ∩ N H \cap N is dense in N N . We show that no compact (or more generally, ω \omega -bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group G G is compact if and only if each continuous homomorphism π : G → H \pi :G \to H of G G onto a topological group H 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 δ {G_\delta } -subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal G δ {G_\delta } -subgroups is torsion. Under Lusin’s hypothesis 2 ω 1 = 2 ω {2^{{\omega _1}}} = {2^\omega } the converse is true for a compact Abelian group G G . If G G is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup K K of G G and a proper, totally dense subgroup H H of G G with K ⊆ H K \subseteq H (in particular, H H is pseudocompact).