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Arhangel'skiĭ sheaf amalgamations in topological groups

Tom 232 / 2016

Boaz Tsaban, Lyubomyr Zdomskyy Fundamenta Mathematicae 232 (2016), 281-293 MSC: 26A03, 03E75. DOI: 10.4064/fm994-1-2016 Opublikowany online: 4 January 2016

Streszczenie

We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property $\alpha _{1.5}$ is equivalent to Arhangel’skiĭ’s formally stronger property $\alpha _1$. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space $X$ such that the space ${\rm C_{p}}(X)$ of continuous real-valued functions on $X$ with the topology of pointwise convergence has Arhangel’skiĭ’s property $\alpha _1$ but is not countably tight. This follows from results of Arhangel’skiĭ–Pytkeev, Moore and Todorčević, and provides a new solution, with stronger properties than the earlier solution, of a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces.

Autorzy

  • Boaz TsabanDepartment of Mathematics
    Bar-Ilan University
    Ramat Gan 5290002, Israel
    and
    Department of Mathematics
    Weizmann Institute of Science
    Rehovot 7610001, Israel
    e-mail
  • Lyubomyr ZdomskyyKurt Gödel Research Center for Mathematical Logic
    University of Vienna
    Währinger Str. 25
    1090 Wien, Austria
    e-mail

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