Three topological reducibilities for discontinuous functions
Abstract
We define a family of three related reducibilities, $\leq_T$, $\leq_{tt}$ and $\leq_m$, for arbitrary functions $f,g:X\rightarrow\mathbb R$, where $X$ is a compact separable metric space. The $\equiv_T$equivalence classes mostly coincide with the proper Baire classes. We show that certain $\alpha$jump functions $j_\alpha:2^\omega\rightarrow \mathbb R$ are $\leq_m$minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $\leq_{tt}$ and $\leq_m$, finding an exact match to the $\alpha$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.07600
 arXiv:
 arXiv:1906.07600
 Bibcode:
 2019arXiv190607600D
 Keywords:

 Mathematics  Logic;
 03D30;
 03E15;
 03D55;
 26A21
 EPrint:
 36 pages