Arhangel'skiĭ sheaf amalgamations in topological groups (intermediate report)

Details Arhangel'skiĭ sheaf amalgamations in topological groups (intermediate report) Arhangel'skiĭ sheaf amalgamations in topological groups (intermediate report)

2011-03-25

arxiv.org/abs/1103.4957v1

Journal of the European Mathematical Society 12 (2012), 353-372

Mathematics

General Topology

Dedicated to Dikran Dikranjan, on the occasion of his 60-th birthday. The next version is expected to include examples disting

Scientific paper

10.4171/JEMS/305

The main results in the present version of this paper are: 1. For arbitrary topological groups, Nyikos's property $\alpha_{1.5}$ is equivalent to Arhangel'ski\u{\i}'s $\alpha_1$. 2. There is a topological space $X$ such that $C_p(X)$ is $\alpha_1$ but is not countably tight. Item (1) solves a problem of Shakhmatov (2002), and is proved using a new kind of perturbation argument. Item (2), which is proved by quoting known results (of Arhangel'ski\u{\i}-Pytkeev, Moore and Todor\v{c}evi\'c), gives a new solution, with remarkable properties, to a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces. This problem was first solved by Plichko (2009) using Banach spaces with weaker locally convex topologies.

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