Skip to main content
SpringerLink
Log in

Menger remainders of topological groups

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is \(\sigma \)-compact. Also, the existence of a Scheepers non-\(\sigma \)-compact remainder of a topological group follows from CH and yields a P-point, and hence is independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel’skii.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arhangel’skii, A.: Hurewicz spaces, analytic sets and fan tightness of function spaces. Doklady Akademii Nauk SSSR 287, 525–528 (1986). (In Russian)

    MathSciNet  Google Scholar 

  2. Arhangel’skii, A., van Mill, J.: Topological homogeneity. In: Hart, K.P., van Mill, J., Simon, P. (eds.) Recent Progress in General Topology III, pp. 1–68. Springer, Berlin (2014)

    Chapter  Google Scholar 

  3. Arhangel’skii, A., Tkachenko, M.: Topological Groups and Related Structures. Atlantis Studies in Mathematics, 1. Atlantis Press, World Scientific Publishing Co. Pte. Ltd., Paris, Hackensack (2008)

  4. Arhangel’skii, A.: Two types of remainders of topological groups. Comment. Math. Univ. Carolin. 49, 119–126 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Arhangel’skii, A.: The Baire property in remainders of topological groups and other results. Comment. Math. Univ. Carolin. 50, 273–279 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Aurichi, L.F.: \(D\)-Spaces, topological games, and selection principles. Topol. Proc. 36, 107–122 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Aurichi, L.F., Bella, A.: When is a space Menger at infinity? Appl. Gen. Topol. 16, 75–80 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Banakh, T., Zdomskyy, L.: Selection principles and infinite games on multicovered spaces. In: Kocinac, L.D.R. (ed.) Selection Principles and Covering Properties in Topology, Quaderni di Matematica 18, Dept. Math., Seconda Universita di Napoli, Caserta, pp. 1–51 (2006)

  9. Blass, A., Hrušák, M., Verner, J.: On strong P-points. Proc. Am. Math. Soc. 141, 2875–2883 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canjar, R.M.: Mathias forcing which does not add dominating reals. Proc. Am. Math. Soc. 104, 1239–1248 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chodounsky, D., Repovš, D., Zdomskyy, L.: Mathias forcing and combinatorial covering properties of filters. J. Symb. Logic 80, 1398–1410 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Debs, G.: Espaces hrditairement de Baire. (French) [Hereditarily Baire spaces]. Fundam. Math. 129, 199–206 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Engelking, R.: General Topology. Monografie Matematyczne, Vol. 60. PWN—Polish Scientific Publishers, Warsaw (1977)

  14. Henriksen, M., Isbell, J.R.: Some properties of compactifications. Duke Math. J. 25, 83–105 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hrušák, M., Minami, H.: Laver–Prikry and Mathias–Prikry type forcings. Ann. Pure Appl. Logic 165, 880–894 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hurewicz, W.: Über die Verallgemeinerung des Borellschen Theorems. Math. Z. 24, 401–421 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  17. Just, W., Miller, A.W., Scheepers, M., Szeptycki, P.J.: The combinatorics of open covers II. Topol. Appl. 73, 241–266 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, 156. Springer, New York (1995)

    Book  Google Scholar 

  19. Repovš, D., Semenov, P.: Continuous Selections of Multivalued Mappings. Mathematics and Its Applications, vol. 455. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  20. Repovš, D., Zdomskyy, L., Zhang, S.: Countable dense homogeneous filters and the Menger covering property. Fundam. Math. 224, 233–240 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Samet, N., Scheepers, M., Tsaban, B.: Partition relations for Hurewicz-type selection hypotheses. Topol. Appl. 156, 616–623 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Scheepers, M.: Combinatorics of open covers. I. Ramsey theory. Topol. Appl. 69, 31–62 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shelah, S.: Proper and Improper Forcing. Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  24. Tkachuk, V.: A Cp-Theory Problem Book: Special Features of Function Spaces. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  25. Tsaban, B.: Menger’s and Hurewicz’s problems: solutions from “the book” and refinements. In: Set Theory and Its Applications. Contemporary Mathematics, vol. 533, pp. 211–226. American Mathematical Society, Providence (2011)

  26. Tsaban, B., Zdomskyy, L.: Combinatorial images of sets of reals and semifilter trichotomy. J. Symb. Logic 73, 1278–1288 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zdomskyy, L.: \(o\)-Boundedness of free objects over a Tychonoff space. Mat. Stud. 25, 10–28 (2006)

    MathSciNet  Google Scholar 

  28. Zdomskyy, L.: A semifilter approach to selection principles. Comment. Math. Univ. Carolin. 46, 525–539 (2005)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Catania, Cittá universitaria, viale A. Doria 6, 95125, Catania, Italy

    Angelo Bella

  2. Department of Mathematics, Faculty of Science, Hacettepe University, 06800, Beytepe–Ankara, Turkey

    Seçil Tokgöz

  3. Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, 1090, Vienna, Austria

    Lyubomyr Zdomskyy

Authors
  1. Angelo Bella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seçil Tokgöz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lyubomyr Zdomskyy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lyubomyr Zdomskyy.

Additional information

The research of Angelo Bella that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM. Seçil Tokgöz would like to thank Hacettepe University BAP project 014 G 602 002 for its support. Lyubomyr Zdomskyy would like to thank the Austrian Academy of Sciences (APART Program) as well as the Austrian Science Fund FWF (Grant I 1209-N25) for generous support for this research.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bella, A., Tokgöz, S. & Zdomskyy, L. Menger remainders of topological groups. Arch. Math. Logic 55, 767–784 (2016). https://doi.org/10.1007/s00153-016-0493-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-016-0493-8

Keywords

Mathematics Subject Classification

Search

Navigation