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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 \(\omega_1\).

03E35 Consistency and independence results
22A05 Structure of general topological groups
54E35 Metric spaces, metrizability
Full Text: DOI
[1] Proceedings of the American Mathematical Society 108 pp 241– (1990)
[2] Algebra, logic, set theory. Festschrift für Ulrich Felgner zum 65. Geburtstag 4 pp 63– (2007)
[3] DOI: 10.1016/j.apal.2005.07.003 · Zbl 1102.03047
[4] Russian Mathematical Surveys 36 pp 151– (1981)
[5] DOI: 10.1016/S0166-8641(01)00125-0 · Zbl 1008.54002
[6] Moscow University Mathematics Bulletin 54 pp 33– (1999)
[7] DOI: 10.1016/0166-8641(82)90065-7 · Zbl 0503.54020
[8] DOI: 10.1016/0166-8641(92)90021-Q · Zbl 0774.54019
[9] DOI: 10.1016/B978-044452208-5/50023-5
[10] DOI: 10.1007/s00153-008-0104-4 · Zbl 1171.03029
[11] Ordering MAD families a la Katětov 68 pp 1337– (2003)
[12] DOI: 10.1016/j.topol.2005.11.021 · Zbl 1133.54002
[13] DOI: 10.1016/j.topol.2003.09.014 · Zbl 1085.54016
[14] DOI: 10.1111/j.1749-6632.1989.tb22391.x
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