Wyniki wyszukiwania

1

On the closure of Baire classes under transfinite convergences

100%

Mátrai T.

Fundamenta Mathematicae

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2004

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tom 183

|

nr 2

157-168

EN

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.

2

Weak difference property of functions with the Baire property

100%

Mátrai T.

Fundamenta Mathematicae

|

2003

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tom 177

|

nr 1

1-17

EN

We prove that the class of functions with the Baire property has the weak difference property in category sense. That is, every function for which f(x+h) - f(x) has the Baire property for every h ∈ ℝ can be written in the form f = g + H + ϕ where g has the Baire property, H is additive, and for every h ∈ ℝ we have ϕ(x+h) - ϕ (x) ≠ 0 only on a meager set. We also discuss the weak difference property of some subclasses of the class of functions with the Baire property, and the consistency of the difference property of the class of functions with the Baire property.

3

On the difference property of Borel measurable functions

64%

Fujita H., Mátrai T.

Fundamenta Mathematicae

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2010

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tom 208

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nr 1

57-73

EN

If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.

4

On splitting infinite-fold covers

51%

Elekes M., Mátrai T., Soukup L.

Fundamenta Mathematicae

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2011

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tom 212

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nr 2

95-127

EN

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.

5

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

51%

Coskey S., Mátrai T., Steprāns J.

Fundamenta Mathematicae

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2013

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tom 223

|

nr 1

29-48

EN

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality 𝔭 ≤ 𝔟 does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we use our results to give an embedding from the inclusion ordering on 𝒫(ω) into the Borel Tukey ordering on cardinal invariants.

Ograniczanie wyników

5 Fundamenta Mathematicae

5 Mátrai T.

1 Coskey S.

1 Elekes M.

1 Fujita H.

1 Soukup L.

1 Steprāns J.

1 2013

1 2011

1 2010

1 2004

1 2003

On the closure of Baire classes under transfinite convergences

Weak difference property of functions with the Baire property

On the difference property of Borel measurable functions

On splitting infinite-fold covers

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum