Decomposing Borel functions using the Shore–Slaman join theorem
Tom 230 / 2015
Streszczenie
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.
