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2 Fundamenta Mathematicae

2 2016

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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.

The linear refinement number and selection theory

80%

Machura M. , Shelah S. , Tsaban B.

nr 1

15-40

The linear refinement number 𝔩𝔯 is the minimal cardinality of a centered family in $[ω]^{ω}$ such that no linearly ordered set in $([ω]^{ω},⊆ *)$ refines this family. The linear excluded middle number 𝔩𝔵 is a variation of 𝔩𝔯. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that 𝔩𝔯 = 𝔩𝔵 = 𝔡 in all models where the continuum is at most ℵ₂, and that the cofinality of 𝔩𝔯 is uncountable. Using the method of forcing, we show that 𝔩𝔯 and 𝔩𝔵 are not provably equal to 𝔡, and rule out several potential bounds on these numbers. Our results solve a number of open problems.