Abstract
We prove that if an ultrafilter \({\mathcal{L}}\) is not coherent to a Q-point, then each analytic non-σ-bounded topological group G admits an increasing chain \({\langle G_\alpha:\alpha < \mathfrak b(\mathcal L)\rangle}\) of its proper subgroups such that: (i) \({\bigcup_{\alpha}G_\alpha=G}\); and (ii) For every σ-bounded subgroup H of G there exists α such that \({H\subset G_\alpha}\). In case of the group Sym(ω) of all permutations of ω with the topology inherited from ω ω this improves upon earlier results of S. Thomas.
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This research was supported in part by the Slovenian Research Agency grants P1-0292-0101, J1-2057-0101 and BI-UA/09-10-005. The third author acknowledges the support of the FWF grant P19898-N18. We thank the referees for comments and suggestions.
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Banakh, T., Repovš, D. & Zdomskyy, L. On the length of chains of proper subgroups covering a topological group. Arch. Math. Logic 50, 411–421 (2011). https://doi.org/10.1007/s00153-010-0222-7
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DOI: https://doi.org/10.1007/s00153-010-0222-7