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On convergent sequences in dual groups

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Abstract

We provide some characterizations of precompact abelian groups G whose dual group \(G_p^\wedge \) 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 \(G_p^\wedge \) by the following property of G: No infinite Hausdorff quotient group of G is countable. Also, we present an example of a dense subgroup G of the compact metrizable group \({\mathbb {Z}}(2)^\omega \) such that G is of the first category in itself, has measure zero, but the dual group \(G_p^\wedge \) does not contain infinite compact subsets. This complements a result of J.E. Hart and K. Kunen (2005) on convergent sequences in dual groups. Making use of the group G we construct the first example of a precompact Pontryagin reflexive abelian group which is of the first Baire category.

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Acknowledgements

The authors are grateful to the anonymous referees for careful reading of the original version of the article and helpful comments.

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Correspondence to S. Hernández.

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M. V. Ferrer was partially supported by the Generalitat Valenciana, Grant GV/2018/110. S. Hernández was partially supported by the Spanish Ministerio de Economía y Competitividad, Grant MTM2016-77143-P (AEI/FEDER, EU). M. Tkachenko: the article was finished during the visit of the third listed author to the Universitat Jaume I, Spain, in June, 2019. He expresses his gratitude to the hosts for financial support and kind attention.

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Ferrer, M.V., Hernández, S. & Tkachenko, M. On convergent sequences in dual groups. RACSAM 114, 71 (2020). https://doi.org/10.1007/s13398-020-00790-x

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Keywords

  • Reflexive
  • Precompact
  • Pseudocompact
  • Baire property
  • Convergent sequence

Mathematics Subject Classification

  • Primary 43A40
  • 22D35
  • Secondary 22C05
  • 54E52
  • 54C10