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

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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).

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Notes

  1. 1.

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

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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.

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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). https://doi.org/10.1007/s13398-020-00793-8

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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