Journal of Mathematical Sciences

, Volume 211, Issue 1, pp 127–135 | Cite as

Countable Powers of Compact Abelian Groups in the Uniform Topology and Cardinality of Their Dual Groups

  • D. Dikranjan
  • E. Martin-Peinador
  • V. Tarieladze


For a topological Abelian group X, we consider in the group X the uniform topology and study some properties of the obtained topological group. In particular, we show, that if \( X\kern0.5em =\kern0.5em \mathbb{S} \) is the circle group, then the group \( {\mathbb{S}}^{\mathbb{N}} \) endowed with the uniform topology has the dual group of cardinality 2c.


Abelian Group Topological Vector Space Dual Group Compact Abelian Group Topological Abelian Group 
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  1. 1.
    A. V. Arhangel’skii and M. G. Tkachenko, Topological Groups and Related Structures, Atlantis Series in Math., Vol. 1, Atlantis Press, World Scientific, Paris–Amsterdam (2008).Google Scholar
  2. 2.
    H. Anzai and S. Kakutani, “Bohr compactifications of a locally compact Abelian group, II,” Proc. Imp. Acad. Tokyo, 19, 533–539 (1943).MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Aussenhofer, “Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups,” Disser. Math., 384, Warsaw (1999).Google Scholar
  4. 4.
    W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lect. Notes Math., 1466, Springer-Verlag, Berlin (1991).Google Scholar
  5. 5.
    N. Bourbaki, Éléments de Mathématique. Topologie Générale. Chapitre X: Espaces Fonctionels, Hermann, Paris (1974).Google Scholar
  6. 6.
    M. J. Chasco, E. Martín-Peinador, and V. Tarieladze, “On Mackey topology for groups,” Stud. Math., 132, No. 3, 257–284 (1999).Google Scholar
  7. 7.
    W. W. Comfort and K. A. Ross, “Pseudocompactness and uniform continuity in topological groups,” Pac. J. Math., 16, 483–496 (1966).MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Dikranjan, E. Martín-Peinador, and V. Tarieladze, “Group valued null sequences and metrizable non-Mackey groups,” in: Forum Math., published online 2012.02.03.Google Scholar
  9. 9.
    D. Dikranjan, Iv. Prodanov, and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure Appl. Math., 130, Marcel Dekker, New York–Basel (1989).Google Scholar
  10. 10.
    R. Engelking, General Topology, Panstwowe Wydawnictwo Naukowe, Warszawa (1985).Google Scholar
  11. 11.
    G. Fichtenhollz and L. Kantorovitch, “Sur les operations lineaires dans l’espace ses fonctions bornees,” Stud. Math., 5, 69–98 (1934).Google Scholar
  12. 12.
    S. S. Gabriyelyan, “Groups of quasi-invariance and the Pontryagin duality,” Topol. Appl., 157, 2786–2802 (2010).MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer-Verlag, Berlin (1963).Google Scholar
  14. 14.
    S. Kakutani, “On cardinal numbers related with a compact Abelian group,” Proc. Imp. Acad. Tokyo, 19, 366–372 (1943).MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    J. W. Nienhuys, “A solenoidal and monothetic minimally almost periodic group,” Fund. Math., 73, No. 2, 167–169 (1971/72).Google Scholar
  16. 16.
    S. Rolewicz, “Some remarks on monothetic groups,” Colloq. Math., 13, 28–29 (1964).Google Scholar
  17. 17.
    L. J. Sulley, “On countable inductive limits of locally compact Abelian groups,” J. London Math. Soc., 5, 629–637 (1972).MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    N. Ya. Vilenkin, “The theory of characters of topological Abelian groups with boundedness given,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, 439–162 (1951).Google Scholar
  19. 19.
    A. Weil, L’intégration dans les groupes topologiques et ses applications, Hermann, Paris (1940).Google Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  • D. Dikranjan
    • 1
  • E. Martin-Peinador
    • 2
  • V. Tarieladze
    • 3
  1. 1.Università di UdineUdineItaly
  2. 2.Universidad Complutense de MadridMadridSpain
  3. 3.N. Muskhelishvili Institute of Computational Mathematics of the Georgian Technical UniversityTbilisiGeorgia

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