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2 Fundamenta Mathematicae

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On the closure of Baire classes under transfinite convergences

100%

Mátrai T.

Fundamenta Mathematicae

2004

tom 183

nr 2

157-168

Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_{α}: X → Y~(α<ω₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set ${α < ω₁: f_{α}(x) ≠ f(x)}$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ⁰_{η}$ sets which can be interesting in its own right.

On splitting infinite-fold covers

51%

Elekes M. , Mátrai T. , Soukup L.

nr 2

95-127

Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on problems with κ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.