Pseudocompact topological group refinements of maximal weight

a b s t r a c tWe show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ♯).Every pseudocompact Abelian group G with cardinality |G| ≤ 2 2 c satisfies this inequality and therefore admits a pseudocompact group topology with property ♯. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property ♯.We also observe that pseudocompact Abelian groups with property ♯ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.

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“…Theorem 5.3 (Theorem 4.5 of [4]). Let G = (G, T 1 ) be a pseudocompact Abelian group with w(G) = α > ω, and set…”

Section: Pseudocompact Groups With Property ♯mentioning

confidence: 99%

Pseudocompact group topologies with no infinite compact subsets

Galindo

Macario

2011

Journal of Pure and Applied Algebra

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“…is independent and has cardinality 2 c , we may construct as in Lemma 4.4 of [9] a collection of sets Z B,C ⊂ X B,C , such that:…”

Section: Some Extension Resultsmentioning

confidence: 99%

Reflexivity in precompact groups and extensions

Galindo

Tkachenko

Bruguera

et al. 2014

Topology and its Applications

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We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that:; (1) A precompact Abelian group G of bounded order is reflexive if the dual group G<^> has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G.; (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive.; We show on the other hand that an extension of a compact group by a reflexive omega-bounded group (even dual to a reflexive P-group) can fail to be reflexive.; We also show that the P-modification of a reflexive sigma-compact group can be non-reflexive (even if, as proved in [20], the P-modification of a locally compact Abelian group is always reflexive). (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (published version

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“…By using Theorem 3.1, the result of Kiltinen was independently improved in 1982 by S. Berhanu, Wis and J. D. Reid [6], and the second author [77] [20], the Peter-Weyl theorem and unitary representations are applied. The number of pseudocompact group topologies on compact Abelian groups is studied in [19] and in [12], with the help of Theorem 3.1. For arbitrary compact groups see [22], Sections 5.2 and 8.2.…”

Section: Theorem 32 For Every Group G There Exists a Functionmentioning

confidence: 99%