Free locally convex spaces with a small base


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.

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  1. 1.

    Arhangel’skii, A.V., Tkachenko, M.G.: Topological groups and related strutures. Atlantis Press/World Scientific, Amsterdam/Raris (2008)

  2. 2.

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

  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)

  5. 5.

    Christensen, J.P.R.: Topology and Borel structure. North-Holland Mathematics Studies, vol. 10, North-Holland, Amsterdam (1974)

  6. 6.

    Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

    Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Ferrando, J.C.: Private communication (2016)

  9. 9.

    Ferrando, J.C., Ka̧kol, J.: On precompact sets in spaces \(C_{c}(X)\). Georgian Math. J. 20, 247–254 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Ferrando, J.C., Ka̧kol, J.: Stronger metrizable locally convex topologies on \(C_{p}(X)\). Preprint

  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)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Flood, J.: Free locally convex spaces. Dissertationes Math CCXXI, PWN, Warczawa (1984)

  13. 13.

    Floret, K.: Weakly Compact Sets. Lecture Notes in Mathematics, vol. 801. Springer, Berlin (1980)

    Google Scholar 

  14. 14.

    Gabriyelyan, S.: The \(k\)-space property for free locally convex spaces. Can. Math. Bull. 57, 803–809 (2014)

    Article  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Gabriyelyan, S., Ka̧kol, J.: On topological spaces and topological groups with certain local countable networks. Topol. Appl. 190, 59–73 (2015)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Gabriyelyan, S., Ka̧kol, J., Leiderman, A.: On topological groups with a small base and metrizability. Fund. Math. 229, 129–158 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

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

    Google 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)

    MathSciNet  Google Scholar 

  23. 23.

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

    Google Scholar 

  24. 24.

    Pol, R.: Normality in function spaces. Fund. Math. 84, 145–155 (1974)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Raĭkov, D.A.: Free locally convex spaces for uniform spaces. Math. Sb. 63, 582–590 (1964)

    MathSciNet  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Uspenskiĭ, V.V.: Free topological groups of metrizable spaces. Math. USSR Izv. 37, 657–680 (1991)

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Saak Gabriyelyan.

Additional information

The research was supported for Jerzy Ka̧kol by Generalitat Valenciana, Conselleria d’Educació i Esport, Spain, Grant PROMETEO/2015/058 and by the GAČR project 16-34860L and RVO: 67985840. Jerzy Ka̧kol gratefully acknowledges also the financial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit March 15–22, 2016.

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Gabriyelyan, S., Ka̧kol, J. Free locally convex spaces with a small base. RACSAM 111, 575–585 (2017).

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