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Decomposing Borel functions using the Shore-Slaman join theorem

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Kihara T.

Fundamenta Mathematicae

2015

tom 230

nr 1

1-13

Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals α and β with α ≤ β < α·2, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class γ functions with $Δ⁰_{β+1}$ domains such that γ + α ≤ β if and only if the preimage of each $Σ^{0}_{α+1}$ set under that function is $Σ^{0}_{β+1}$, and the transformation of a $Σ^{0}_{α+1}$ set into the $Σ^{0}_{β+1}$ preimage is continuous.

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1 Fundamenta Mathematicae

1 Kihara T.

1 2015

Wyniki wyszukiwania

Decomposing Borel functions using the Shore-Slaman join theorem