Compactlike totally dense subgroups of compact groups
Authors:
Dikran N. Dikranjan and Dmitrii B. Shakhmatov
Journal:
Proc. Amer. Math. Soc. 114 (1992), 11191129
MSC:
Primary 22C05; Secondary 22A05
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:
http://dx.doi.org/10.1090/S00029939199210816942
PII:
S 00029939(1992)10816942
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
