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The linear refinement number and selection theory

Michał Machura;  Saharon Shelah;  Boaz Tsaban — 2016

Fundamenta Mathematicae

The linear refinement number 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 is a variation of . 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 = = in all models where the continuum is at most ℵ₂, and that the cofinality of is...

On the dimension of the space of ℝ-places of certain rational function fields

Taras Banakh;  Yaroslav Kholyavka;  Oles Potyatynyk;  Michał Machura;  Katarzyna Kuhlmann — 2014

Open Mathematics

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.

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