Turing degrees in Polish spaces and decomposability of Borel functions [article]

Vassilios Gregoriades, Takayuki Kihara, Keng Meng Ng
2016 arXiv   pre-print
We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative
more » ... results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture, and we explore the idea of applying the technique of turning Borel-measurable functions into continuous ones.
arXiv:1410.1052v2 fatcat:rxyszurelbgg5l7p7awfojh5km