Compact-like totally dense subgroups of compact groups
Dikranjan, Dikran N., 1950- ; Shakhmatov, Dmitri
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 ▫$H \cap N$▫ 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 ▫$\pi :G \to H$▫ 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 ▫$K \subseteq H$▫ (in particular, ▫$H$▫ is pseudocompact).
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