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The Hurewicz covering property and slaloms in the Baire space

100%
Tsaban B.
Fundamenta Mathematicae
|
2004
|
tom 181
|
nr 3
273-280
EN
According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This solves in the affirmative a problem of Scheepers. The proof uses a rigorously justified abuse of notation and a "structure" counterpart of a combinatorial characterization, in terms of slaloms, of the minimal cardinality 𝔟 of an unbounded family of functions in the Baire space. In particular, we obtain a new characterization of 𝔟.
2
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Arhangel'skiĭ sheaf amalgamations in topological groups

63%
Tsaban B., Zdomskyy L.
Fundamenta Mathematicae
|
2016
|
tom 232
|
nr 3
281-293
EN
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.
3
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The linear refinement number and selection theory

51%
Machura M., Shelah S., Tsaban B.
Fundamenta Mathematicae
|
2016
|
tom 234
|
nr 1
15-40
EN
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.
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