Decomposing Borel functions using the Shore–Slaman join theorem

Takayuki Kihara¹

¹ School of Information Science Japan Advanced Institute of Science and Technology 1-1 Asahidai Nomi, Ishikawa, 923-1292 Japan

Fundamenta Mathematicae, Tome 230 (2015) no. 1, pp. 1-13.

Résumé

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.

Détail

DOI : 10.4064/fm230-1-1

Keywords: jayne rogers proved every function analytic space separable metrizable space decomposable countably many continuous functions closed domains only preimage each sigma set under function again sigma many researchers conjectured jayne rogers theorem generalized finite levels borel functions paper using shore slaman join theorem turing degrees following variant jayne rogers theorem finite transfinite levels hierarchy borel functions countable ordinals alpha beta alpha leq beta alpha cdot every function between polish spaces having small transfinite inductive dimension decomposable countably many baire class gamma functions mathbf delta beta domains gamma alpha leq beta only preimage each sigma alpha set under function sigma beta transformation sigma alpha set sigma beta preimage continuous