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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
##### Keywords:
countable Fréchet-Urysohn group
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##### References:
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