Decomposing Borel functions using the Shore-Slaman join theorem [article]

Takayuki Kihara
2013 arXiv   pre-print
Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each F_σ set under it 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
more » ... nite 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 Δ^0_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.
arXiv:1304.0698v1 fatcat:khw75v6g2vglpnc2f6g5u5hnaq