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


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. 1.
    Bernstein, A.R.: A new kind of compactness for topological spaces. Fund. Math. 66, 185–193 (1970)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boero, A.C., Tomita, A.H.: A countably compact group topology on Abelian almost torsion-free groups from selective ultrafilters. Houston J. Math. 39, 317–342 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Castro-Pereira, I., Tomita, A.H.: Abelian torsion groups with a countably compact group topology. Topology Appl. 157, 44–52 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Springer-Verlag (1974)Google Scholar
  5. 5.
    Garcia-Ferreira, S., Tomita, A.H., Watson, S.: Countably compact groups from a selective ultrafilter. Proc. Amer. Math. Soc. 133, 937–943 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Koszmider, P.B., Tomita, A.H., Watson, S.: Forcing countably compact group topologies on a larger free Abelian group. Topology Proc. 25, 563–574 (2000)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kunen, K.: Set Theory: an Introduction to Independence Proofs. North Holland (1983)Google Scholar
  8. 8.
    Madariaga-Garcia, R.E., Tomita, A.H.: Countably compact topological group topologies on free Abelian groups from selective ultrafilters. Topology Appl. 154, 1470–1480 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Numakura, K.: On bicompact semigroups. Math. J. Okayama Univ. 1, 99–108 (1952)MathSciNetzbMATHGoogle Scholar
  10. 10.
    D. Robbie and S. Svetlichny, An answer to A. D. Wallace's question about countably compact cancellative semigroups, Proc. Amer. Math. Soc., 124 (1996), 325–330Google Scholar
  11. 11.
    Shelah, S.: Proper and Improper Forcing. Springer (1998)Google Scholar
  12. 12.
    M. G. Tkachenko, Countably compact and pseudocompact topologies on free Abelian groups, Soviet Math. (Izv. VUZ), 34 (1990), 79–86Google Scholar
  13. 13.
    Tkachenko, M.G., Yaschenko, I.: Independent group topologies on Abelian groups. Topology Appl. 122, 425–451 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tomita, A.H.: The Wallace problem: a counterexample from \({\rm MA}{_{\rm countable}}\) and \(p\)-compactness. Canad. Math. Bull. 39, 486–498 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tomita, A.H.: The existence of initially \(\omega _{1}\)-compact group topologies on free Abelian groups is independent of ZFC. Comment. Math. Univ. Carolinae 39, 401–413 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Tomita, A.H.: A solution to Comfort's question on the countable compactness of powers of a topological group. Fund. Math. 186, 1–24 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Tomita, A.H.: A group topology on the free abelian group of cardinality \({\mathfrak{c}}\) that makes its finite powers countably compact. Topology Appl. 196, 976–998 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tomita, A.H., Watson, S.: Ultraproducts, \(p\)-limits and antichains on the Comfort group order. Topology Appl. 143, 147–157 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wallace, A.D.: The structure of topological semigroups. Bull. Amer. Math. Soc. 61, 95–112 (1955)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations