Decomposing Baire functions

Cichoń, Jacek; Morayne, Michał; Pawlikowski, Janusz; Solecki, Sławomir

doi:10.2307/2275474

Cited by 25 publications

(37 citation statements)

References 4 publications

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“…It is easy to see that for each > 0 the set of points at which the oscillation 6 of E * is greater than must be a closed discrete set. Hence E * is of Baire class 1 on its domain.…”

Section: Definition 22 a Set E ⊆ Rmentioning

confidence: 99%

“…In [6] the authors consider the following question: If A and B are two families of functions between Polish spaces, what is the least cardinal κ such that every member of A can be decomposed into κ members of B -this cardinal κ the authors call dec (A, B). The most natural class to consider for B is the class of continuous functions and the problem to which it gives rise had been posed by Lusin, who wondered whether every Borel function could be decomposed in countably many continuous functions.…”

Section: Decomposing Continuous Functionsmentioning

confidence: 99%

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Decomposing Euclidean space with a small number of smooth sets

Steprāns

1999

Trans. Amer. Math. Soc.

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Abstract. Let the cardinal invariant sn denote the least number of continuously smooth n-dimensional surfaces into which (n + 1)-dimensional Euclidean space can be decomposed. It will be shown to be consistent that sn is greater than s n+1 . These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.

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“…It is easy to see that for each > 0 the set of points at which the oscillation 6 of E * is greater than must be a closed discrete set. Hence E * is of Baire class 1 on its domain.…”

Section: Definition 22 a Set E ⊆ Rmentioning

confidence: 99%

Section: Decomposing Continuous Functionsmentioning

confidence: 99%

Decomposing Euclidean space with a small number of smooth sets

Steprāns

1999

Trans. Amer. Math. Soc.

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“…We will write Π κ for Π κ (R). In [4] the authors considered the following cardinal decomposition function for arbitrary families F ⊂ R Z , with Z ⊂ R, and G ⊂ {R X : X ⊂ Z}:…”

Section: Preliminariesmentioning

confidence: 99%

“…In [4] the authors considered the values of dec(B β , B α ) for α < β < ω 1 , where B α stands for the functions of α-th Baire class. In particular, they proved that where cov(M) is the smallest cardinality of a covering of R by meager sets, and d, the dominating number, is the smallest cardinality of a dominating family D ⊂ ω ω , that is, such that for every f ∈ ω ω there exists g ∈ D such that f ≤ * g. Moreover, in papers [12] and [11] it has been proved that each of these inequalities can be strict.…”

Section: Preliminariesmentioning

confidence: 99%

Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions

Ciesielski¹

1999

Proc. Amer. Math. Soc.

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Abstract. In this paper we will investigate the smallest cardinal number κ such that for any symmetrically continuous function f : R → R there is a partition {X ξ : ξ < κ} of R such that every restriction f X ξ : X ξ → R is continuous. The similar numbers for the classes of Sierpiński-Zygmund functions and all functions from R to R are also investigated and it is proved that all these numbers are equal. We also show that cf(c) ≤ κ ≤ c and that it is consistent with ZFC that each of these inequalities is strict.

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“…Laczkovich showed that for any 1 ≤ β < ω 1 there is an f ∈ L β with dec(f, B β ) > ω (see [CM] for a proof); and by [CMPS,Theorem 5.5] there is an f ∈ L 1 that is f lower semicontinous, with dec(f, B 1 ) ≥ cov (M). The next corollary improves on these results.…”

Section: S Lawomir Soleckimentioning

confidence: 99%

Decomposing Borel sets and functions and the structure of Baire class 1 functions

Solecki

1998

J. Amer. Math. Soc.

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We establish dichotomy results concerning the structure of Baire class 1 functions. We consider decompositions of Baire class 1 functions into continuous functions and into continuous functions with closed domains. Dichotomy results for both of them are proved: a Baire class 1 function decomposes into countably many countinuous functions, or else contains a function which turns out to be as complicated with respect to the decomposition as any other Baire class 1 function; similarly for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprāns. Their proofs use effective descriptive set theory as well as infinite Borel games on the integers. An important role in the proofs is played by what we call, in analogy with being Wadge complete, complete semicontinuous functions. As another application of our study of complete semicontinuous functions, we generalize some recent theorems of Jackson and Mauldin, and van Mill and Pol concerning measures viewed as examples of complicated semicontinuous functions. We also prove that a Borel set A A is either Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } or there is a continuous injection ϕ : ω ω → A \phi :\; \omega ^{\omega }\to A such that for any Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } set B ⊂ A B\subset A , ϕ − 1 ( B ) \phi ^{-1}(B) is meager. We show analogous results for Borel functions. These theorems give a new proof of a result of Stern, strengthen some results of Laczkovich, and improve the estimates for cardinal coefficients studied by Cichoń, Morayne, Pawlikowski, and the author.

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Decomposing Baire functions

Abstract: We discuss in the paper the following problem: Given a function in a given Baire class, into “how many” (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

Cited by 25 publications

References 4 publications

Decomposing Euclidean space with a small number of smooth sets

Decomposing Euclidean space with a small number of smooth sets

Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions

Decomposing Borel sets and functions and the structure of Baire class 1 functions

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