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Decomposing Borel functions using the Shore–Slaman join theorem

Volume 230 / 2015

Takayuki Kihara Fundamenta Mathematicae 230 (2015), 1-13 MSC: Primary 03E15; Secondary 54H05. DOI: 10.4064/fm230-1-1

Abstract

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_\sigma $ set under that function is again $F_\sigma $. 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 $\alpha $ and $\beta $ with $\alpha \leq \beta <\alpha \cdot 2$, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class $\gamma $ functions with $\mathbf {\Delta }^0_{\beta +1}$ domains such that $\gamma +\alpha \leq \beta $ if and only if the preimage of each $\Sigma ^0_{\alpha +1}$ set under that function is $\Sigma ^0_{\beta +1}$, and the transformation of a $\Sigma ^0_{\alpha +1}$ set into the $\Sigma ^0_{\beta +1}$ preimage is continuous.

Authors

  • Takayuki KiharaSchool of Information Science
    Japan Advanced Institute of Science and Technology
    1-1 Asahidai
    Nomi, Ishikawa, 923-1292 Japan
    e-mail

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