Countable tightness and \({\mathfrak {G}}\)-bases on free topological groups

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Given a Tychonoff space X, let F(X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. In this paper, we consider two topological properties of F(X) or A(X), namely the countable tightness and \(\mathfrak G\)-base. We provide some characterizations of the countable tightness and \(\mathfrak G\)-base of F(X) and A(X) for various special classes of spaces X. Furthermore, we also study the countable tightness and \(\mathfrak G\)-base of some \(F_{n}(X)\) of F(X).

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  1. 1.

    Recently, T. Banakh has told me that he has given an affirmative answer to this question.


  1. 1.

    Arhangel’skiǐ, A.V.: Hurewicz spaces, analytic sets and fan-tightness of spaces of functions. Sov. Math. Dokl. 33(2), 396–399 (1986)

  2. 2.

    Arhangel’skiǐ, A.V., Bella, A.: Countable fan-tightness versus countable tightness. Comment. Math. Univ. Carol. 37(3), 567–578 (1996)

  3. 3.

    Arhangel’skiǐ, A.V., Tkachenko, M.G.: Topological Groups and Related Structures. Atlantis Press and World Scientific, Paris (2008)

  4. 4.

    Arhangel’skiǐ, A.V., Okunev, O.G., Pestov, V.G.: Free topological groups over metrizable spaces. Topol. Appl. 33, 63–76 (1989)

  5. 5.

    Banakh, T.: \(\mathfrak{P}_{0}\)-spaces. Topol. Appl. 195, 151–173 (2015)

  6. 6.

    Banakh, T., Leiderman, A.: The strong Pytkeev property in topological spaces. Topol. Appl. 227, 10–29 (2017)

  7. 7.

    Banakh, T.: The strong Pytkeev\(^{\ast }\) property of topological spaces (2019). arXiv:1607.03599v3

  8. 8.

    Banakh, T.: Topological spaces with an \(\omega ^{\omega }\)-base (2019). arXiv:1607.07978v10

  9. 9.

    Cai, Z., Lin, S.: Sequentially compact spaces with a point-countable \(k\)-network. Topol. Appl. 193, 162–166 (2015)

  10. 10.

    Chis, C., Vincenta Ferrer, M., Hernández, S., Tsaban, B.: The character of topological groups, via bounded systems, Pontryagin-van Kampen duality and pcf theory. J. Algebra 420, 86–119 (2014)

  11. 11.

    Dudley, R.M.: Continuity of homomorphisms. Duke Math. J. 28, 587–594 (1961)

  12. 12.

    Engelking, R.: General Topology (revised and completed edn.). Heldermann, Berlin (1989)

  13. 13.

    Ferrando, J.C., Ka̧kol, J., López Pellicer, M., Saxon, S.A.: Tightness and distinguished Fréchet spaces. J. Math. Anal. Appl. 324, 862–881 (2006)

  14. 14.

    Fletcher, P., Lindgren, W.F.: Quasi-uniform Spaces. Marcel Dekker, New York (1982)

  15. 15.

    Frolík, Z.: Generalizations of the \(G\)-property of complete metric spaces. Czech. Math. J. 10, 359–379 (1960)

  16. 16.

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

  17. 17.

    Gabriyelyan, S.S., Ka̧kol, J., Leiderman, A.: The strong Pytkeev property for topological groups and topological vector spaces. Monatsh Math. 175, 519–542 (2014)

  18. 18.

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

  19. 19.

    Gabriyelyan, S.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)

  20. 20.

    Gabriyelyan, S.S., Ka̧kol, J.: On \({\mathfrak{B}} \)-space and related concepts. Topol. Appl. 191, 178–198 (2015)

  21. 21.

    Graev, M.I.: Free topological groups. In: Topology and Topological Algebra, Translations Series 1, vol. 8, pp. 305–364. American Mathematical Society (1962) [Russian original in: Izvestiya Akad. Nauk SSSR Ser. Mat., 12, 279–323 (1948)]

  22. 22.

    Gruenhage, G.: Generalized metric spaces. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 423–501. Elsevier Science Publishers B.V, Amsterdam (1984)

  23. 23.

    Gruenhage, G., Michael, E.A., Tanaka, Y.: Spaces determined by point-countable covvers. Pac. J. Math. 113, 303–332 (1984)

  24. 24.

    Gruenhage, G., Tanaka, Y.: Products of \(k\)-spaces and spaces of countable tightness. Trans. Am. Math. Soc. 273, 299–308 (1982)

  25. 25.

    Guthrie, J.A.: A characterization of \(\aleph _{0}\)-spaces. Gen. Topol. Appl. 1, 105–110 (1971)

  26. 26.

    Kanatani, Y., Sasaki, N., Nagata, J.: New characterizations of some generalized metric spaces. Math. Jpn. 30, 805–820 (1985)

  27. 27.

    Leiderman, A.G., Pestov, V.G., Tomita, A.H.: On topological groups admitting a base at indentity indexed with \(\omega ^\omega \). Fund. Math. (2015). arXiv:1511.07062v1(accepted)

  28. 28.

    Li, Z., Lin, F., Liu, C.: Networks on free topological groups. Topol. Appl. 180, 186–198 (2015)

  29. 29.

    Lin, F., Liu, C., Cao, J.: wo weak forms of countability axioms in free topological groups. Topol. Appl. 207, 96–108 (2016)

  30. 30.

    Lin, S., Tanaka, Y.: Point-countable \(k\)-networks, closed maps, and related results. Topol. Appl. 59, 79–86 (1994)

  31. 31.

    Markov, A.A.: On free topological groups. Izv. Akad. Nauk SSSR Ser. Mat. 9, 3–64 (1945) [Amer. Math. Soc. Transl., 8, 195–272 (1962); (in Russian)]

  32. 32.

    Nickolas, P., Tkachenko, M.: Local compactness in free topological groups. Bull. Aust. Math. Soc. 68(2), 243–265 (2003)

  33. 33.

    O’Meara, P.: On paracompactness in function spaces with the compact-open topology. Proc. Am. Math. Soc. 29, 183–189 (1971)

  34. 34.

    Pytkeev, E.G.: Maximally decomposable spaces. Trudy Mat. Inst. Steklov. 154, 209–213 (1983)

  35. 35.

    Sipacheva, O.V.: Free topological groups of spaces and their subspaces. Topol. Appl. 101, 181–212 (2000)

  36. 36.

    Šneǐder, V.: Continuous images of Souslin and Borel sets; metrization theorems. Dokl. Acad. Nauk USSR 50, 77–79 (1945)

  37. 37.

    Tkachenko, M.G.: On a spectral decomposition of free topological groups. Usp. Mat. Nauk 39(2), 191–192 (1984)

  38. 38.

    Boaz Tsaban, L.: Zdomskyy, On the Pytkeev property in spaces of continuous functions (II). Houst. J. Math. 35, 563–571 (2009)

  39. 39.

    Yamada, K.: Characterizations of a metrizable space such that every \(A_n(X)\) is a \(k\)-space. Topol. Appl. 49, 74–94 (1993)

  40. 40.

    Yamada, K.: Tightness of free Abelian topological groups and of finite product of sequntial fans. Topol. Proc. 22, 363–381 (1997)

  41. 41.

    Yamada, K.: Metrizable subspaces of free topological groups on metrizable spaces. Topol. Proc. 23, 379–409 (1998)

  42. 42.

    Yamada, K.: The natural mappings \(i_{n}\) and \(k\)-subspaces of free topological groups on metrizable spaces. Topol. Appl. 146–147, 239–251 (2005)

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The authors wish to thank professors Salvador Hernández and Boaz Tsaban for telling us some information of the paper [10]. Moreover, the authors wish to thank professor Chuan Liu for reading parts of this paper and making comments. Finally, we hope to thank professor Shou Lin for finding a gap in our proof of Theorem 3.18 in the previous version and giving some key for us to supplement the proof.

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Correspondence to Fucai Lin.

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Fucai Lin is supported by the NSFC (No. 11571158), the Natural Science Foundation of Fujian Province (No. 2017J01405) of China, the Program for New Century Excellent Talents in Fujian Province University, the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Date Science and Statistics.

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Lin, F., Ravsky, A. & Zhang, J. Countable tightness and \({\mathfrak {G}}\)-bases on free topological groups. RACSAM 114, 67 (2020).

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  • Free topological group
  • Free Abelian topological group
  • Countable tightness
  • Countable fan-tightness
  • \({\mathfrak {G}}\)-base
  • strong Pytkeev property
  • Universally uniform \({\mathfrak {G}}\)-base

Mathematics Subject Classification

  • Primary 54H11
  • 22A05
  • Secondary 54E20
  • 54E35
  • 54D50
  • 54D55