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## Fundamenta Mathematicae

2015 | 230 | 1 | 1-13
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### Decomposing Borel functions using the Shore-Slaman join theorem

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EN
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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Tom
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1-13
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wydano
2015
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autor
• School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292 Japan
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