Arhangel'skiĭ sheaf amalgamations in topological groups

Volume 232 / 2016

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


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.


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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image