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Arhangel’skiĭ sheaf amalgamations in topological groups. (English) Zbl 1348.54033

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.

MSC:

54H11 Topological groups (topological aspects)
03E75 Applications of set theory
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54C35 Function spaces in general topology
26A03 Foundations: limits and generalizations, elementary topology of the line
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