Decomposing Borel functions using the Shore-Slaman join theorem
Takayuki Kihara · arXiv (Cornell University) · 2013
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 finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ and $β$ with $α\leqβ