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Countably compact group topologies on the free Abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter

  • A. C. BoeroEmail author
  • I. C. Pereira
  • A. H. Tomita
Article

Abstract

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

Key words and phrases

topological group countable compactness selective ultrafilter free Abelian group Wallace’s problem 

Mathematics Subject Classification

primary 54H11 22A05 secondary 54A35 54G20 

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Notes

Acknowledgement

The authors would like to thank Vinicius de Oliveira Rodrigues for revising and suggesting improvements in the last version of this work prior to submission.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Centro de Matemática, Computação e Cognição (CMCC)Universidade Federal do ABC (UFABC)Santo AndréBrazil
  2. 2.Instituto de Ciências Exatas e NaturaisUniversidade Federal do ParáBelémBrazil
  3. 3.Departamento de Matemática, Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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