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

  • Fucai LinEmail author
  • Alex Ravsky
  • Jing Zhang
Original Paper
  • 7 Downloads

Abstract

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).

Keywords

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 

Notes

Acknowledgements

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.

References

  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)Google Scholar
  2. 2.
    Arhangel’skiǐ, A.V., Bella, A.: Countable fan-tightness versus countable tightness. Comment. Math. Univ. Carol. 37(3), 567–578 (1996)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arhangel’skiǐ, A.V., Tkachenko, M.G.: Topological Groups and Related Structures. Atlantis Press and World Scientific, Paris (2008)CrossRefGoogle Scholar
  4. 4.
    Arhangel’skiǐ, A.V., Okunev, O.G., Pestov, V.G.: Free topological groups over metrizable spaces. Topol. Appl. 33, 63–76 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Banakh, T.: \(\mathfrak{P}_{0}\)-spaces. Topol. Appl. 195, 151–173 (2015)CrossRefGoogle Scholar
  6. 6.
    Banakh, T., Leiderman, A.: The strong Pytkeev property in topological spaces. Topol. Appl. 227, 10–29 (2017)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dudley, R.M.: Continuity of homomorphisms. Duke Math. J. 28, 587–594 (1961)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Engelking, R.: General Topology (revised and completed edn.). Heldermann, Berlin (1989)Google Scholar
  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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fletcher, P., Lindgren, W.F.: Quasi-uniform Spaces. Marcel Dekker, New York (1982)zbMATHGoogle Scholar
  15. 15.
    Frolík, Z.: Generalizations of the \(G\)-property of complete metric spaces. Czech. Math. J. 10, 359–379 (1960)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gabriyelyan, S.S., Ka̧kol, J., Leiderman, A.: On topological groups with a small base and metrizability. Fundam. Math. 229, 129–158 (2015)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gabriyelyan, S.S., Ka̧kol, J.: On topological spaces and topological groups with certain local countable networks. Topol. Appl. 190, 59–73 (2015)MathSciNetCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  20. 20.
    Gabriyelyan, S.S., Ka̧kol, J.: On \({\mathfrak{B}} \)-space and related concepts. Topol. Appl. 191, 178–198 (2015)MathSciNetCrossRefGoogle Scholar
  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)] Google Scholar
  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)CrossRefGoogle Scholar
  23. 23.
    Gruenhage, G., Michael, E.A., Tanaka, Y.: Spaces determined by point-countable covvers. Pac. J. Math. 113, 303–332 (1984)CrossRefGoogle Scholar
  24. 24.
    Gruenhage, G., Tanaka, Y.: Products of \(k\)-spaces and spaces of countable tightness. Trans. Am. Math. Soc. 273, 299–308 (1982)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Guthrie, J.A.: A characterization of \(\aleph _{0}\)-spaces. Gen. Topol. Appl. 1, 105–110 (1971)CrossRefGoogle Scholar
  26. 26.
    Kanatani, Y., Sasaki, N., Nagata, J.: New characterizations of some generalized metric spaces. Math. Jpn. 30, 805–820 (1985)MathSciNetzbMATHGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lin, F., Liu, C., Cao, J.: wo weak forms of countability axioms in free topological groups. Topol. Appl. 207, 96–108 (2016)CrossRefGoogle Scholar
  30. 30.
    Lin, S., Tanaka, Y.: Point-countable \(k\)-networks, closed maps, and related results. Topol. Appl. 59, 79–86 (1994)MathSciNetCrossRefGoogle Scholar
  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)] Google Scholar
  32. 32.
    Nickolas, P., Tkachenko, M.: Local compactness in free topological groups. Bull. Aust. Math. Soc. 68(2), 243–265 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    O’Meara, P.: On paracompactness in function spaces with the compact-open topology. Proc. Am. Math. Soc. 29, 183–189 (1971)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Pytkeev, E.G.: Maximally decomposable spaces. Trudy Mat. Inst. Steklov. 154, 209–213 (1983)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Sipacheva, O.V.: Free topological groups of spaces and their subspaces. Topol. Appl. 101, 181–212 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Šneǐder, V.: Continuous images of Souslin and Borel sets; metrization theorems. Dokl. Acad. Nauk USSR 50, 77–79 (1945)MathSciNetGoogle Scholar
  37. 37.
    Tkachenko, M.G.: On a spectral decomposition of free topological groups. Usp. Mat. Nauk 39(2), 191–192 (1984)MathSciNetGoogle Scholar
  38. 38.
    Boaz Tsaban, L.: Zdomskyy, On the Pytkeev property in spaces of continuous functions (II). Houst. J. Math. 35, 563–571 (2009)zbMATHGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Yamada, K.: Tightness of free Abelian topological groups and of finite product of sequntial fans. Topol. Proc. 22, 363–381 (1997)zbMATHGoogle Scholar
  41. 41.
    Yamada, K.: Metrizable subspaces of free topological groups on metrizable spaces. Topol. Proc. 23, 379–409 (1998)MathSciNetzbMATHGoogle Scholar
  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)CrossRefGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2020

Authors and Affiliations

  1. 1.School of mathematics and statisticsMinnan Normal UniversityZhangzhouPeople’s Republic of China
  2. 2.Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of NASULvivUkraine
  3. 3.School of mathematics and statisticsMinnan Normal UniversityZhangzhouPeople’s Republic of China

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