## 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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