Decomposing Borel sets and functions and the structure of Baire class 1 functions

Abstract

We establish dichotomy results concerning the structure of Baire class 1 functions. We consider decompositions of Baire class 1 functions into continuous functions and into continuous functions with closed domains. Dichotomy results for both of them are proved: a Baire class 1 function decomposes into countably many countinuous functions, or else contains a function which turns out to be as complicated with respect to the decomposition as any other Baire class 1 function; similarly for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprāns. Their proofs use effective descriptive set theory as well as infinite Borel games on the integers. An important role in the proofs is played by what we call, in analogy with being Wadge complete, complete semicontinuous functions. As another application of our study of complete semicontinuous functions, we generalize some recent theorems of Jackson and Mauldin, and van Mill and Pol concerning measures viewed as examples of complicated semicontinuous functions. We also prove that a Borel set A A is either Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } or there is a continuous injection ϕ : ω ω → A \phi :\; \omega ^{\omega }\to A such that for any Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } set B ⊂ A B\subset A , ϕ − 1 ( B ) \phi ^{-1}(B) is meager. We show analogous results for Borel functions. These theorems give a new proof of a result of Stern, strengthen some results of Laczkovich, and improve the estimates for cardinal coefficients studied by Cichoń, Morayne, Pawlikowski, and the author.

Playing in the first Baire class

“…Another result, due to Jayne and Rogers, concerns a fragment of the first Baire class . Its 0‐dimensional version states that a function is ‐preserving iff there is a countable partition of A in closed sets such that for each integer i the function is continuous, we say that f is piecewise continuous .…”

On the Structure of Finite Level and ω-Decomposable Borel Functions

“…The Pawlikowski function P is the countable power P = f : ( + 1) → of the function f: + 1 → defined by f(n) = n + 1 if n = and f( ) = 0, where + 1 is endowed with the order topology and with the discrete topology. By [Sol98,Theorem 4.1] (see Theorem 2.4 below), P is in a sense the canonical example of a Baire class 1 function which is not -decomposable into continuous functions.…”

“…More precisely, he proved the following results: 2 The original Jayne-Rogers theorem holds also in the broader context of nonseparable metrizable spaces when X is assumed to be absolute Souslin-F (i.e., the counterpart of an analytic space in the realm of nonseparable spaces). However, for the sake of simplicity, here and in the subsequent weak and strong generalization of the Jayne-Rogers theorem (Conjectures 1.4 and 1.6) we will just consider the already relevant and well-studied case of separable metrizable spaces (see e.g., [Sol98]).…”

Borel-amenable reducibilities for sets of reals

“…Assume that the first alternative does not hold. Since each A n is an analytic subset of a Polish space, by definition it is also Souslin and hence we can apply Solecki's Theorem 4.1 of [12] to / \ A". But by our assumption it can not be the case that the first alternative of Solecki's Theorem holds for each f \ A", thus the second alternative must hold for some index k G co, that is P C / | " A^.…”