Arhangel’skiĭ sheaf amalgamations in topological groups
| Autor: | Boaz Tsaban, Lyubomyr Zdomskyy |
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| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Fundamenta Mathematicae. :1-13 |
| ISSN: | 1730-6329 0016-2736 |
| DOI: | 10.4064/fm994-1-2016 |
| Popis: | 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'ski\u{\i}'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 $C_p(X)$ of continuous real-valued functions on $X$, with the topology of pointwise convergence, has Arhangel'ski\u{\i}'s property $\alpha_1$ but is not countably tight. This result follows from results of Arhangel'ski\u{\i}--Pytkeev, Moore and Todor\v{c}evi\'c, and provides a new solution, with remarkable properties, to a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces. The Averbukh--Smolyanov problem was first solved by Plichko (2009), using Banach spaces with weaker locally convex topologies. Comment: Final version (minor changes) |
| Databáze: | OpenAIRE |
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