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Archive for Mathematical Logic

, Volume 50, Issue 3–4, pp 411–421 | Cite as

On the length of chains of proper subgroups covering a topological group

  • Taras Banakh
  • Dušan Repovš
  • Lyubomyr Zdomskyy
Original Article

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.

Keywords

Q-points Pκ-point σ-bounded group ω-bounded group Menger property \({[\mathcal{F}]}\)-Menger property 

Mathematics Subject Classification (2000)

Primary: 03E17 54H11 Secondary: 54D20 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Taras Banakh
    • 1
    • 2
  • Dušan Repovš
    • 3
    • 4
  • Lyubomyr Zdomskyy
    • 5
  1. 1.Department of MathematicsIvan Franko National University of LvivLvivUkraine
  2. 2.Instytut MatematykiUniwersytet Humanistyczno-PrzyrodniczyKielcePoland
  3. 3.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  4. 4.Faculty of EducationUniversity of LjubljanaLjubljanaSlovenia
  5. 5.Kurt Gödel Research Center for Mathematical LogicUniversity of ViennaWienAustria

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