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Selective separability properties of Fréchet–Urysohn spaces and their products

Volume 263 / 2023

Serhii Bardyla, Fortunato Maesano, Lyubomyr Zdomskyy Fundamenta Mathematicae 263 (2023), 271-299 MSC: Primary 03E50; Secondary 54D65, 03E35, 54D10, 03E17, 03E65. DOI: 10.4064/fm230522-13-10 Published online: 15 November 2023


We study the behaviour of selective separability properties in the class of Fréchet–Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Fréchet–Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $\mathfrak p=\mathfrak c$, we construct such an example which is also zero-dimensional and $\alpha _{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Fréchet–Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.


  • Serhii BardylaInstitute of Mathematics
    P.J. Šafárik University
    Košice, Slovakia
    Institute of Discrete Mathematics and Geometry
    TU Wien
    Wien, Austria
  • Fortunato MaesanoMIFT - Matematica e Informatica
    Università degli Studi di Messina
    98166 Messina, Italy
  • Lyubomyr ZdomskyyInstitut für Diskrete Mathematik und Geometrie
    Technische Universität Wien
    1040 Wien, Austria

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