Advertisement

Free locally convex spaces with a small base

  • Saak GabriyelyanEmail author
  • Jerzy Ka̧kol
Original Paper

Abstract

The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a \(\mathfrak {G}\)-base. A space X has a \(\mathfrak {G}\)-base if for every \(x\in X\) there is a base \(\{ U_\alpha : \alpha \in \mathbb {N}^\mathbb {N}\}\) of neighborhoods at x such that \(U_\beta \subseteq U_\alpha \) whenever \(\alpha \le \beta \) for all \(\alpha ,\beta \in \mathbb {N}^\mathbb {N}\), where \(\alpha =(\alpha (n))_{n\in \mathbb {N}}\le \beta =(\beta (n))_{n\in \mathbb {N}}\) if \(\alpha (n)\le \beta (n)\) for all \(n\in \mathbb {N}\). We show that if X is an Ascoli \(\sigma \)-compact space, then L(X) has a \(\mathfrak {G}\)-base if and only if X admits an Ascoli uniformity \(\mathcal {U}\) with a \(\mathfrak {G}\)-base. We prove that if X is a \(\sigma \)-compact Ascoli space of \(\mathbb {N}^\mathbb {N}\)-uniformly compact type, then L(X) has a \(\mathfrak {G}\)-base. As an application we show: (1) if X is a metrizable space, then L(X) has a \(\mathfrak {G}\)-base if and only if X is \(\sigma \)-compact, and (2) if X is a countable Ascoli space, then L(X) has a \(\mathfrak {G}\)-base if and only if X has a \(\mathfrak {G}\)-base.

Keywords

Free locally convex space \(\mathfrak {G}\)-base \(C_k(X)\) Compact resolution 

References

  1. 1.
    Arhangel’skii, A.V., Tkachenko, M.G.: Topological groups and related strutures. Atlantis Press/World Scientific, Amsterdam/Raris (2008)Google Scholar
  2. 2.
    Banakh, T., Gabriyelyan, S.: On the Ck-stable closure of the class of (separable) metrizable spaces. Monatshefte Math. 180, 39–64 (2016)Google Scholar
  3. 3.
    Banakh, T., Leiderman, A.: \(\mathfrak{G} \)-bases in free (locally convex) topological vector spaces. arXiv:1606.01967
  4. 4.
    Chis C., Ferrer M. V., Hernádez S., Tsaban B.: The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory, J. Algebra 420, 86–119 (2014)Google Scholar
  5. 5.
    Christensen, J.P.R.: Topology and Borel structure. North-Holland Mathematics Studies, vol. 10, North-Holland, Amsterdam (1974)Google Scholar
  6. 6.
    Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)zbMATHGoogle Scholar
  7. 7.
    Ferrando, J.C.: On uniform spaces with a small base and \(K\)-analytic \(C_c(X)\). Topol. Appl. 193, 77–83 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ferrando, J.C.: Private communication (2016)Google Scholar
  9. 9.
    Ferrando, J.C., Ka̧kol, J.: On precompact sets in spaces \(C_{c}(X)\). Georgian Math. J. 20, 247–254 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ferrando, J.C., Ka̧kol, J.: Stronger metrizable locally convex topologies on \(C_{p}(X)\). PreprintGoogle Scholar
  11. 11.
    Ferrando, J.C., Ka̧kol, J., López Pellicer, M., Saxon, S.A.J.: Tightness and distinguished Fréchet spaces. Math. Anal. Appl. 324, 862–881 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Flood, J.: Free locally convex spaces. Dissertationes Math CCXXI, PWN, Warczawa (1984)Google Scholar
  13. 13.
    Floret, K.: Weakly Compact Sets. Lecture Notes in Mathematics, vol. 801. Springer, Berlin (1980)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gabriyelyan, S.: The \(k\)-space property for free locally convex spaces. Can. Math. Bull. 57, 803–809 (2014)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gabriyelyan, S.: A characterization of free locally convex spaces over metrizable spaces which have countable tightness. Scientiae Mathematicae Japonicae 78, 201–205 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gabriyelyan, S., Ka̧kol, J.: On topological spaces and topological groups with certain local countable networks. Topol. Appl. 190, 59–73 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gabriyelyan, S., Ka̧kol, J., Kubzdela, A., Lopez-Pellicer, M.: On topological properties of Fréchet locally convex spaces with the weak topology. Topol. Appl. 192, 123–137 (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gabriyelyan, S., Ka̧kol, J., Leiderman, A.: The strong Pytkeev property for topological groups and topological vector spaces. Monatsch. Math. 175, 519–542 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gabriyelyan, S., Ka̧kol, J., Leiderman, A.: On topological groups with a small base and metrizability. Fund. Math. 229, 129–158 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kakol, J., Kubiś, W., Lopez-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  21. 21.
    Leiderman, A.G, Pestov, V.G., Tomita, A.H.: On topological groups admitting a base at identity indexed with \(w^w\). arXiv:1511.07062
  22. 22.
    Markov, A.A.: On free topological groups. Dokl. Akad. Nauk SSSR 31, 299–301 (1941)MathSciNetGoogle Scholar
  23. 23.
    McCoy, R.A., Ntantu, I.: Topological Properties of Spaces of Continuous Functions. Lecture Notes in Math., vol. 1315. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  24. 24.
    Pol, R.: Normality in function spaces. Fund. Math. 84, 145–155 (1974)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Raĭkov, D.A.: Free locally convex spaces for uniform spaces. Math. Sb. 63, 582–590 (1964)MathSciNetGoogle Scholar
  26. 26.
    Tkachuk, V.V.: A space \(C_p(X)\) is dominated by irrationals if and only if it is \(K\)-analytic. Acta Math. Hung. 107, 253–265 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Uspenskiĭ, V.V.: Free topological groups of metrizable spaces. Math. USSR Izv. 37, 657–680 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBeershebaIsrael
  2. 2.A. Mickiewicz UniversityPoznanPoland
  3. 3.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic

Personalised recommendations