THREE TOPOLOGICAL REDUCIBILITIES FOR DISCONTINUOUS FUNCTIONS

We define a family of three related reducibilities, ≤_T, ≤_tt and ≤_m, for arbitrary functions f, g: X → R, where X is a compact separable metric space. The ≡_T-equivalence classes mostly coincide with the proper Baire classes. We show that certain α-jump functions j_α: 2^ω → R are ≤_m-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to ≤_tt and ≤_m, finding an exact match to the α hierarchy introduced by Bourgain [Bull. Soc. Math. Belg. Sér. B 32 (1980), pp. 235–249] and analyzed in Kechris & Louveau [Trans. Amer. Math. Soc. 318 (1990), pp. 209–236].