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Countable Fréchet Boolean groups: an independence result. (English) Zbl 1233.03053
The paper is motivated by a known problem of Malykhin on the ZFC-existence of a countable non-metrizable Fréchet-Urysohn group and a related problem of G. Gruenhage and P. J. Szeptycki [Topology Appl. 151, No. 1–3, 238–259 (2005; Zbl 1085.54016)] on the ZFC-existence of a filter \(\mathcal F\) on \(\omega\) such that \(([\omega]^{<\omega},\tau_{\mathcal F})\) is a non-metrizable Fréchet-Urysohn group. Here \([\omega]^{<\omega}\) is the group of all finite subsets of \(\omega\) endowed with the operation of symmetric difference and the group topology \(\tau_{\mathcal F}\) whose neighborhood base at the neutral element consists of the subgroups \([F]^{<\omega}\) where \(F\in\mathcal F\). The main result of the paper is Theorem 1 saying that it is consistent that \(\omega_2=\mathfrak c\) and there is no filter \(\mathcal F\) on \(\omega\) such that the topological group \(([\omega]^{<\omega},\tau_{\mathcal F})\) is Fréchet-Urysohn and is non-metrizable of weight \(\omega_1\).

MSC:
03E35 Consistency and independence results
22A05 Structure of general topological groups
54E35 Metric spaces, metrizability
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