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Compact-like totally dense subgroups of compact groups

Authors: Dikran N. Dikranjan and Dmitrii B. Shakhmatov
Journal: Proc. Amer. Math. Soc. 114 (1992), 1119-1129
MSC: Primary 22C05; Secondary 22A05
MathSciNet review: 1081694
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Abstract: 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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Keywords: Compact group, countably compact group, pseudocompact group, totally minimal group, totally dense subgroup, $ {G_\delta }$-subgroup, Lusin's hypothesis
Article copyright: © Copyright 1992 American Mathematical Society