Pseudocompact group topologies and totally dense subgroups

Comfort, W. W.; Soundararajan, T.

doi:10.2140/pjm.1982.100.61

Cited by 43 publications

(25 citation statements)

References 23 publications

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“…Our first result shows that both "Abelian" and "totally disconnected" are superfluous here. This theorem also substantially strengthens Corollary 6.8 of [5]. It should be noted that a general topological (Abelian) group can contain a proper, totally dense, countably compact subgroup.…”

Section: Resultssupporting

confidence: 72%

“…In 1982 Comfort and Soundararajan proved that any countably compact totally dense subgroup of a totally disconnected compact Abelian group G coincides with G [5,Theorem 6.7]. Our first result shows that both "Abelian" and "totally disconnected" are superfluous here.…”

Section: Resultsmentioning

confidence: 73%

“…The notion of a totally dense subgroup was extensively investigated in the literature [2,3,4,5,6,7,8,12]. Its significance became clear in the context of the open mapping theory in the following sense.…”

Section: Resultsmentioning

confidence: 99%

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Compact-like totally dense subgroups of compact groups

Dikranjan¹,

Shakhmatov²

1992

Proc. Amer. Math. Soc.

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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).

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Section: Resultssupporting

confidence: 72%

Section: Resultsmentioning

confidence: 73%

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Compact-like totally dense subgroups of compact groups

Dikranjan¹,

Shakhmatov²

1992

Proc. Amer. Math. Soc.

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“…Since this topic is a bit removed from our central focus here, for details in this direction we simply refer the reader to the relevant papers known to us: [50,51], [36] [52][53][54][55]. We note explicitly that, building upon and extending results from her thesis [56], Giordano Bruno and Dikranjan [57] characterized those compact abelian groups with a proper totally dense pseudocompact subgroup as those with no closed torsion G δ -subgroup.…”

Section: Totally Dense Subgroupsmentioning

confidence: 99%

Non-Abelian Pseudocompact Groups

Comfort

Remus

2016

Axioms

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“…Comfort and Soundararajan [4] first studied the question of when a compact abelian group K admits a proper totally dense pseudocompact subgroup and provided a solution in the case when K is connected (iff K is not metrizable). Later Comfort and Robertson [2] studied the compact abelian groups K that admit a totally dense pseudocompact subgroup of size < |K|.…”

Section: Introductionmentioning

confidence: 99%

Pseudocompact totally dense subgroups

Dikranjan

Bruno

2007

Proc. Amer. Math. Soc.

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Abstract. It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K admits a proper totally dense pseudocompact subgroup, then K cannot have a torsion closed G δ -subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H of K that contains an ω-bounded dense subgroup of K (such an H is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc.

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Pseudocompact group topologies and totally dense subgroups

Cited by 43 publications

References 23 publications

Compact-like totally dense subgroups of compact groups

Compact-like totally dense subgroups of compact groups

Non-Abelian Pseudocompact Groups

Pseudocompact totally dense subgroups

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