Ukrainian Mathematical Journal

, Volume 67, Issue 12, pp 1922–1931 | Cite as

Ultrafilters on Balleans

  • I. V. Protasov
  • S. V. Slobodianiuk
Article
  • 26 Downloads

A ballean (or, equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), estimate the basic cardinal invariants of the corona, and characterize the subsets of balleans in terms of companions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, R.I. (2003).Google Scholar
  2. 2.
    I. Protasov and T. Banakh, “Ball structures and colorings of groups and graphs,” Math. Stud. Monogr. Ser., 11 (2003).Google Scholar
  3. 3.
    I. Protasov and M. Zarichnyi, “General asymptology,” Math. Stud. Monogr. Ser., Vol. 12 (2007).Google Scholar
  4. 4.
    I. V. Protasov, “Normal ball structures,” Math. Stud., 20, 3–16 (2003).MathSciNetMATHGoogle Scholar
  5. 5.
    O. Petrenko, I. V. Protasov, and S. Slobodianiuk, “Asymptotic structures of cardinals,” Appl. Gen. Topol., 15, 137–146 (2014).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    I. V. Protasov and A. Tsvietkova, “Decomposition of cellular balleans,” Topology Proc., 36, 1–7 (2010).MathSciNetMATHGoogle Scholar
  7. 7.
    T. Banakh, I. Protasov, D. Repovs, and S. Slobodianiuk, Classifying Homogeneous Cellular Ordinal Balleans, Preprint (http:axiv. org/abs/1409. 3910).Google Scholar
  8. 8.
    O. Petrenko and I. V. Protasov, “Balleans and G-spaces,” Ukr. Math. J., 64, No. 3, 344–350 (2012).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    I. V. Protasov, “Coronas of balleans,” Topol. Appl., 146, 149–161 (2005).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    I. V. Protasov, “Ultrafilters on metric spaces,” Topol. Appl., 164, 207–214 (2014).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    I. Protasov and S. Slobodianiuk, “Ultracompanions of subsets of groups,” Comment. Math. Univ. Carol., 55, 257–265 (2014).MathSciNetMATHGoogle Scholar
  12. 12.
    T. Banakh, I. Protasov, and S. Slobodianiuk, “Scattered subsets of groups,” Ukr. Math. J., 67, No. 3, 304–312 (2015).MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Protasov and S. Slobodianiuk, “On the subset combinatorics of G-spaces,” Algebra Discrete Math., 15, No. 1, 98–109 (2014).MathSciNetMATHGoogle Scholar
  14. 14.
    I. V. Protasov, “Cellularity and density of balleans,” Appl. Gen. Topol., 8, 283–291 (2007).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    M. Filali and I. V. Protasov, “Spread of balleans,” Appl. Gen. Topol., 9, 169–175 (2008).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    J. Baumgartner, “Almost-disjoint sets, the dense set problem, and the partition calculus,” Ann. Math. Logic., 9, 401–439 (1976).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    I. V. Protasov, “Coronas of ultrametric spaces,” Comment. Math. Univ. Carol., 52, 303–307 (2011).MathSciNetMATHGoogle Scholar
  18. 18.
    T. Banakh, O. Chervak, and L. Zdomsky, “On character of points in the Higson corona of a metric space,” Comment. Math. Univ. Carol., 50, 159–178 (2013).MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • I. V. Protasov
    • 1
  • S. V. Slobodianiuk
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations