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

  • T. Banakh
  • S. Gabriyelyan


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 



The authors are deeply indebted to Professor R. Pol for fruitful discussion on the Ascoli property in function spaces. We would like to thank the referee for valuable remarks and suggestions.


  1. 1.
    Banakh, T.: On topological groups containing a Frechet-Urysohn fan. Mat. Stud. 9, 149–154 (1998)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banakh, T.: \(\mathfrak{P}_{0}\)-spaces. Topol. Appl. 195, 151–173 (2015)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Engelking, R.: General topology. Heldermann Verlag, Berlin (1989)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gabriyelyan, S.: Topologies on groups determined by sets of convergent sequences. J. Pure Appl. Algebra 217, 786–802 (2013)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Noble, N.: Ascoli theorems and the exponential map. Trans. Am. Math. Soc. 143, 393–411 (1969)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

Personalised recommendations