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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1081694-2

MathSciNet review:
1081694

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Abstract: A subgroup of a topological group is (weakly) totally dense in if for each closed (normal) subgroup of the set is dense in . We show that no compact (or more generally, -bounded) group contains a proper, totally dense, countably compact subgroup. This yields that a countably compact Abelian group is compact if and only if each continuous homomorphism of onto a topological group 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 -subgroup. If a topological group contains a proper, totally dense, pseudocompact subgroup, then none of its closed, normal -subgroups is torsion. Under Lusin's hypothesis the converse is true for a compact Abelian group . If is a compact Abelian group with nonmetrizable connected component of zero, then there are a dense, countably compact subgroup of and a proper, totally dense subgroup of with (in particular, is pseudocompact).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1081694-2

Keywords:
Compact group,
countably compact group,
pseudocompact group,
totally minimal group,
totally dense subgroup,
-subgroup,
Lusin's hypothesis

Article copyright:
© Copyright 1992
American Mathematical Society