Turing degrees in Polish spaces and decomposability of Borel functions

Vassilios Gregoriades; Takayuki Kihara; Keng Meng Ng

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arXiv · 1410.1052

Turing degrees in Polish spaces and decomposability of Borel functions

Vassilios Gregoriades·Takayuki Kihara·Keng Meng Ng

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Abstract

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 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.

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BibTeX RIS

Vassilios Gregoriades, Takayuki Kihara, Keng Meng Ng. 2016-05-26. Turing degrees in Polish spaces and decomposability of Borel functions. https://arxiv.org/abs/1410.1052

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