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2 Fundamenta Mathematicae

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Arhangel'skiĭ sheaf amalgamations in topological groups

100%

Tsaban B. , Zdomskyy L.

nr 3

281-293

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 $α_{1.5}$ is equivalent to Arhangel'skiĭ's formally stronger property α₁. 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'skiĭ's property α₁ but is not countably tight. This follows from results of Arhangel'skiĭ-Pytkeev, Moore and Todorčević, and provides a new solution, with stronger properties than the earlier solution, of a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces.

Δ₁-Definability of the non-stationary ideal at successor cardinals

80%

Friedman S.-D. , Wu L. , Zdomskyy L.

nr 3

231-254

Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in $L^{ℙ}$ the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).