Three topological reducibilities for discontinuous functions

Abstract

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 [Bou80] and analyzed in Kechris-Louveau [KL90].

Topological reducibilities for discontinuous functions and their structures

“…The latter notion is introduced by Bourgain [4] to prove a refinement of the Odell-Rosenthal theorem in Banach space theory: The 1 -index of a separable Banach space is related to the degrees of discontinuity (the Bourgain rank) of double-dual elements as Baire-one functions. Day-Downey-Westrick [6] showed the following:…”

“…Recently, Day-Downey-Westrick [6] introduced a "many-one"-like ordering ≤ m on real-valued functions on Cantor space. Their ordering ≤ m measures the topological complexity of sets separating the lower level sets from the upper level sets of a function.…”

“…Definition 1.1 (Day-Downey-Westrick [6]). For f, g : 2 ω → R, we say that f is mreducible to g (written f ≤ m g) if for any rationals p and ε > 0, there are rationals r and δ > 0 and a continuous function θ : 2 ω → 2 ω such that, for any x ∈ 2 ω , g(θ(x)) < r + δ implies f (x) < p + ε, and g(θ(x)) > r − δ implies f (x) > p − ε.…”

“…Fortunately, Pequignot [16] has overcome this difficulty by modifying the definition of Wadge reducibility using the theory of an admissible representation, and then, showed that the modified Wadge degree structure of subsets of a secondcountable space is semi-well-ordered. Day-Downey-Westrick [6] adopted a similar idea to consider the notion of m-reducibility for functions of compact metrizable domain.…”

“…We discuss the Bourgain rank in non-compact spaces (which is also considered by Elekes-Kiss-Vidnyánszky [9] via the change of topology, in order to generalize the notion of ranks to Baire class ξ functions, and then to study a cardinal invariant associated with systems of difference equations). Then, based on the notion of sidedness conditions introduced by Day-Downey-Westrick [6], we can classify real-valued functions on a (possibly non-compact) Polish space into (ordered) 5 types (see Definition 4.4). We then generalize the main result in [6] to arbitrary Polish domains as follows: Let X and Y be Polish spaces and let f : X → R and g : Y → R be Baire-one functions.…”