Arhangel'skiĭ sheaf amalgamations in topological groups

Boaz Tsaban¹ ; Lyubomyr Zdomskyy²

¹ Department of Mathematics Bar-Ilan University Ramat Gan 5290002, Israel and Department of Mathematics Weizmann Institute of Science Rehovot 7610001, Israel
² Kurt Gödel Research Center for Mathematical Logic University of Vienna Währinger Str. 25 1090 Wien, Austria

Fundamenta Mathematicae, Tome 232 (2016) no. 3, pp. 281-293.

Résumé

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.

Détail

DOI : 10.4064/fm994-1-2016

Keywords: consider amalgamation properties convergent sequences topological groups topological vector spaces main result paper arbitrary topological groups nyikos property alpha equivalent arhangel ski formally stronger property alpha result solves problem shakhmatov its proof uses perturbation argument prove there topological space space continuous real valued functions topology pointwise convergence has arhangel ski property alpha countably tight follows results arhangel ski pytkeev moore todor evi provides solution stronger properties earlier solution problem averbukh smolyanov concerning topological vector spaces