Abstract
We consider the following problem: For which Tychonoff topological spaces X do the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial (i.e. not finitely generated) metrizable quotient group? First, we give a complete solution of this problem for the key class of compact spaces X. Then, relying on this result, we resolve the problem for several more general important classes of spaces X, including the class of \(\sigma \)-compact spaces, the class of pseudocompact spaces, the class of \(\omega \)-bounded spaces, the class of Čech-complete spaces and the class of K-analytic spaces. Also we describe all absolutely analytic metric spaces X (in particular, completely metrizable spaces) such that the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial metrizable quotient group. Our results are based on an extensive use of topological properties of non-scattered spaces.
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Acknowledgements
The authors thank the referees for advices and several useful remarks. The second listed author gratefully acknowledges the financial support and hospitality received from the Center for Advanced Studies in Mathematics of the Ben-Gurion University of the Negev during his visit in May, 2019.
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Leiderman, A., Tkachenko, M. Metrizable quotients of free topological groups. RACSAM 114, 124 (2020). https://doi.org/10.1007/s13398-020-00855-x
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DOI: https://doi.org/10.1007/s13398-020-00855-x