Precompact noncompact reflexive Abelian groups

Ardanza-Trevijano, Sergio; Chasco, M.J.; Domínguez, Xabier; Tkachenko, Mikhail

doi:10.1515/form.2011.061

Cited by 18 publications

(22 citation statements)

References 16 publications

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“…The following result is very useful for constructing examples of reflexive groups. It follows from [8, Lemma 2.3 and Theorem 6.1] and can be found explicitly in [1,Theorem 2.8].…”

Section: Applications Of the Main Resultsmentioning

confidence: 99%

The impact of the Bohr topology on selective pseudocompactness

Shakhmatov

Yáñez

2019

Topology and its Applications

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Recall that a space X is selectively pseudocompact if for every sequence {Un : n ∈ N} of non-empty subsets of X one can choose a point xn ∈ Un for all n ∈ N such that the resulting sequence {xn : n ∈ N} has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

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“…The following result is very useful for constructing examples of reflexive groups. It follows from [8, Lemma 2.3 and Theorem 6.1] and can be found explicitly in [1,Theorem 2.8].…”

Section: Applications Of the Main Resultsmentioning

confidence: 99%

The impact of the Bohr topology on selective pseudocompactness

Shakhmatov

Yáñez

2019

Topology and its Applications

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“…In the fall of 1990's several specialists in the Pontryagin duality theory were discussing the problem whether every precompact reflexive abelian group was compact. Counterexamples to this conjecture appeared in [2] and [15], where the authors established the existence of pseudocompact noncompact reflexive abelian groups. Clearly every pseudocompact space has the Baire property.…”

Section: Compact Subsets Of Dual Groupsmentioning

confidence: 99%

On convergent sequences in dual groups

Ferrer

Hernández

Tkachenko

2020

RACSAM

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We provide some characterizations of precompact abelian groups G whose dual group G ∧ p endowed with the pointwise convergence topology on elements of G contains a nontrivial convergent sequence. In the special case of precompact abelian torsion groups G, we characterize the existence of a nontrivial convergent sequence in G ∧ p by the following property of G: No infinite quotient group of G is countable. Finally, we present an example of a dense subgroup G of the compact metrizable group Z(2) ω such that G is of the first category in itself, has measure zero, but the dual group G ∧ p does not contain infinite compact subsets. This complements Theorem 1.6 in [J.E. Hart and K. Kunen, Limits in function spaces and compact groups, Topol. Appl. 151 (2005), 157-168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.

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“…Taking into account Proposition 2.8 (1), in what follows we identify (algebraically) F 0 (X) ∧ with its image in (X ∧ ) N and say that an…”

Section: Some Properties Of the Dual Group Of F 0 (X)mentioning

confidence: 99%

“…It is known (see [1,6]), that there are reflexive precompact non-compact Abelian groups. So it is natural to ask: Problem 5.2.…”

Section: Open Questionsmentioning

confidence: 99%