Arhangel'skiĭ sheaf amalgamations in topological groups (Q2787128)

The authors consider amalgamation properties \(\alpha_i\) for \(i=1\), \(1.5\), \(2\), \(3\), \(4\),~\(5\) of convergence of sequences in topological groups (due to Arhangelskiĭ except~\(\alpha_{1.5}\) which is due to Nyikos). They prove that a~topological group is \(\alpha_{1.5}\) if and only if it is~\(\alpha_1\) which solves a~question of Shakhmatov. As an application they get a~short proof of a~theorem of Nogura and Shakhmatov stating that every \(\alpha_{1.5}\)~topological group is Ramsey. Applying results of Arhangel'skiĭ-Pytkeev, Moore and Todorčević they prove that there is a Fréchet-Urysohn \(L\)-space~\(L\) such that \(C_p(L)\) is \(\alpha_1\), it is not Fréchet-Urysohn, it is not countably tight, and every separable subspace of \(C_p(L)\) is metrizable. This is another solution of a~problem of Averbukh and Smolyanov whether every \(\alpha_1\)~topological vector space is Fréchet-Urysohn.