Topological properties of function spaces over ordinals


A topological space X is said to be an Ascoli space if any compact subset \(\mathcal {K}\) of \(C_k(Y)\) is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the \(k_\mathbb {R}\)-property and the Ascoli property of \(C_{p}(\kappa )\) and \(C_k(\kappa )\) over ordinals \(\kappa \). We prove that \(C_p(\kappa )\) is always an Ascoli space, while \(C_p(\kappa )\) is a \(k_\mathbb {R}\)-space iff the cofinality of \(\kappa \) is countable. In particular, this provides the first \(C_{p}\)-example of an Ascoli space which is not a \(k_\mathbb {R}\)-space, namely \(C_p(\omega _1)\). We show that \(C_k(\kappa )\) is Ascoli iff \(\mathrm {cf}(\kappa )\) is countable iff \(C_k(\kappa )\) is metrizable.

This is a preview of subscription content, access via your institution.


  1. 1.

    Arhangel’skii A. V.: Topological Function Spaces. Math. Appl., vol. 78. Kluwer Academic, Dordrecht (1992)

  2. 2.

    Arhangel’skii, A.V.: Normality and dense subspaces. Proc. Am. Math. Soc. 48, 283–291 (2001)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Banakh, T., Gabriyelyan, S.: On the \(C_k\)-stable closure of the class of (separable) metrizable spaces. Monatshefte Math. 180, 39–64 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Engelking, R.: General Topology. Panstwowe Wydawnictwo Naukowe, Waszawa (1977)

    MATH  Google Scholar 

  5. 5.

    Gabriyelyan, S., Grebík, J., Ka̧kol, J., Zdomskyy, L.: The Ascoli property for function spaces. Topol. Appl. 214, 35–50 (2016)

  6. 6.

    Gabriyelyan, S., Ka̧kol, J., Plebanek, G.: The Ascoli property for function spaces and the weak topology on Banach and Fréchet spaces. Stud. Math. 233, 119–139 (2016)

    MATH  Google Scholar 

  7. 7.

    Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, New York (1960)

    Book  MATH  Google Scholar 

  8. 8.

    Gul’ko, S.P.: Spaces of continuous functions on ordinals and ultrafilters. Math. Notes 47, 329–334 (1990)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ka̧kol, J., Kubiś, W., Lopez-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics. Springer, New York (2011)

  10. 10.

    McCoy, R.A., Ntantu, I.: Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics, vol. 1315. Springer, Berlin, Heidelberg (1988)

  11. 11.

    Morris, P.D., Wulbert, D.E.: Functional representation of topological algebras. Pac. J. Math. 22, 323–337 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Sakai, M.: Two properties of \(C_p(X)\) weaker than Fréchet-Urysohn property. Topol. Appl. 153, 2795–2804 (2006)

    Article  MATH  Google Scholar 

  13. 13.

    Tkachuk, V.: A \(C_p\)-theory problem book, special features of function spaces. Springer, New York (2014)

    MATH  Google Scholar 

  14. 14.

    Wilansky, A.: Mazur spaces. Int. J. Math. 4, 39–53 (1981)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Saak Gabriyelyan.

Additional information

The second author was supported by the GACR project 15-34700L and RVO: 67985840. The third author was supported by Generalitat Valenciana, Conselleria d’Educació, Cultura i Esport, Spain, Grant PROMETEO/2013/058 and by the GAČR project 16-34860L and RVO: 67985840, and gratefully acknowledges also the financial support he received from the Kurt Goedel Research Center in Wien for his research visit in days 15.04–24.04 2016. The fourth author would like to thank the Austrian Science Fund FWF (Grant I 1209-N25) for generous support for this research. The collaboration of the second and the fourth authors was partially supported by the Czech Ministry of Education Grant 7AMB15AT035 and RVO: 67985840.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gabriyelyan, S., Grebík, J., Ka̧kol, J. et al. Topological properties of function spaces over ordinals. RACSAM 111, 1157–1161 (2017).

Download citation


  • \(C_p(X)\)
  • \(C_k(X)\)
  • Ascoli
  • \(k_\mathbb {R}\)-space
  • Ordinal space

Mathematics Subject Classification

  • 54C35
  • 54F05
  • 46A08
  • 54E18