Monatshefte für Mathematik

, Volume 180, Issue 1, pp 39–64 | Cite as

On the \(C_k\)-stable closure of the class of (separable) metrizable spaces



Denote by \(\mathbf {C}_k[\mathfrak M]\) the \(C_k\)-stable closure of the class \(\mathfrak M\) of all metrizable spaces, i.e., \(\mathbf {C}_k[\mathfrak M]\) is the smallest class of topological spaces that contains \(\mathfrak M\) and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces \(C_k(X,Y)\) with Lindelöf domain in this class. We show that the class \(\mathbf {C}_k[\mathfrak M]\) coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces \(C_k(X,Y)\) with a separable metrizable space X and a metrizable space Y. We say that a topological space Z is Ascoli if every compact subset of \(C_k(Z)\) is evenly continuous; by the Ascoli Theorem, each k-space is Ascoli. We prove that the class \(\mathbf {C}_k[\mathfrak M]\) properly contains the class of all Ascoli \(\aleph _0\)-spaces and is properly contained in the class of \(\mathfrak {P}\)-spaces, recently introduced by Gabriyelyan and Kąkol. Consequently, an Ascoli space Z embeds into the function space \(C_k(X,Y)\) for suitable separable metrizable spaces X and Y if and only if Z is an \(\aleph _0\)-space.


Metric space Function space Ascoli space \(\aleph _0\)-space \(\mathfrak {P}\)-space \(C_k\)-stable closure 

Mathematics Subject Classification

46E10 54C35 54E18 


  1. 1.
    Banakh, T.: On topological groups containing a Frechet-Urysohn fan. Mat. Stud. 9, 149–154 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Banakh, T.: Topologies on groups determined by sequences: answers to several questions of I. Protasov and E. Zelenyuk. Mat. Stud. 2(15), 145–150 (2001)MathSciNetMATHGoogle Scholar
  3. 3.
    Banakh, T.: \(\mathfrak{P}_{0}\)-spaces. Topol. Appl. 195, 151–173 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Banakh T.: Generalizations of \(k\)-spaces and their applications in general topology, function spaces, and Banach space theory (2015, preprint)Google Scholar
  5. 5.
    Banakh T., Gabriyelyan S.: The \(C_p\)-stable closure of the class of separable metrizable spaces. arXiv:1412.2240
  6. 6.
    Borges, C.: On stratifiable spaces. Pac. J. Math. 17, 1–16 (1966)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Engelking, R.: General topology. Heldermann Verlag, Berlin (1989)MATHGoogle Scholar
  8. 8.
    Ferrando, J.C., Ka̧kol, J.: On precompact sets in spaces \(C_{c}\left( X\right) \). Georgian Math. J. 20, 247–254 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Foged, L.: Characterizations of \(\aleph \)-spaces. Pac. J. Math. 110, 59–63 (1984)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gabriyelyan, S.: Topologies on groups determined by sets of convergent sequences. J. Pure Appl. Algebra 217, 786–802 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gabriyelyan S.: Topological properties of function spaces \({C_k} (X,2)\). arXiv:1504.04198
  12. 12.
    Gabriyelyan, S., Kąkol, J.: On \(\mathfrak{P}\)-spaces and related concepts. Topol. Appl. 191, 178–198 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gabriyelyan, S., Kąkol, J., Kubiś, W., Marciszewski, W.: Networks for the weak topology of Banach and Fréchet spaces. J. Math. Anal. Appl. 432, 1183–1199 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gabriyelyan S., Kąkol J., Plebanek G.: The ascoli property for function spaces and the weak topology of Banach and Fréchet spaces. arXiv:1504.04202
  15. 15.
    Gruenhage G.: Generalized metric spaces. Handbook of set-theoretic topology, pp. 423–501. North-Holland, Amsterdam (1984)Google Scholar
  16. 16.
    Kechris, A.: Classical descriptive set theory. Springer-Verlag, New York (1995)CrossRefMATHGoogle Scholar
  17. 17.
    McCoy R.A., Ntantu I.: Topological properties of spaces of continuous functions. Lecture Notes in Math, vol. 1315 (1988)Google Scholar
  18. 18.
    O’Meara, P.: On paracompactness in function spaces with the compact-open topology. Proc. Am. Math. Soc. 29, 183–189 (1971)MathSciNetMATHGoogle Scholar
  19. 19.
    Michael, E.: \(\aleph _{0}\)-spaces. J. Math. Mech. 15, 983–1002 (1966)MathSciNetGoogle Scholar
  20. 20.
    Michael, E.: On \(k\)-spaces, \(k_{R}\)-spaces and \(k(X)\). Pac. J. Math. 47, 487–498 (1973)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Noble, N.: Ascoli theorems and the exponential map. Trans. Am. Math. Soc. 143, 393–411 (1969)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pol R.: A remark on a question of T. Banakh and S. Gabriyelyan, handwritten notes (2015)Google Scholar
  23. 23.
    Reznichenko, E.: Stratifiability of \(C_k(X)\) for a class of separable metrizable \(X\). Topol. Appl. 155, 2060–2062 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Ivan Franko National UniversityL’vivUkraine
  2. 2.Jan Kochanowski UniversityKielcePoland
  3. 3.Department of MathematicsBen-Gurion University of the NegevBeer ShevaIsrael

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