Topological groups which satisfy an open mapping theorem

Grant, Douglass L.

doi:10.2140/pjm.1977.68.411

Cited by 36 publications

(16 citation statements)

References 9 publications

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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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Section: Totally Dense Subgroupsmentioning

confidence: 99%

Non-Abelian Pseudocompact Groups

Comfort

Remus

2016

Axioms

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“…Hence the quotient group G/K is topologically isomorphic to the image ϕ(G) = A(ω + 1) (see [23,Lemma 1.3]). Since the group A(ω + 1) is countable, all pseudocompact subspaces of A(ω + 1) are compact and metrizable.…”

Section: Remark 42mentioning

confidence: 99%

The three space problem in topological groups

Bruguera¹,

Tkachenko²

2006

Topology and its Applications

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We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p = c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups. (M. Bruguera), mich@xanum.uam.mx (M. Tkachenko).

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“…The concept has been studied in its own right [46], [24], [13], [39], and in connection with the problem of characterizing those topological groups (G, &~) for which ^is minimal among (Hausdorff) topologies making G a topological group [47], [23].…”

Section: W W Comfort and Lewis C Robertsonmentioning

confidence: 99%