Uniformity results on the Baire property

TL;DR

This paper investigates the definability of witnessing the Baire property for Borel and projective sets, showing that continuous functions suffice for almost all codes in the projective case, but not for all codes.

Contribution

It demonstrates the possibility of using continuous functions to witness the Baire property for most projective codes and establishes limitations for all codes, analyzing the complexity within the Borel hierarchy.

Findings

Continuous functions can witness the Baire property for almost all projective codes.

It is impossible to do so for all codes with more complex functions.

Provides complexity estimates for functions verifying the Baire property across the Borel hierarchy.

Abstract

We are concerned with the problem of witnessing the Baire property of the Borel and the projective sets (assuming determinacy) through a sufficiently definable function in the codes. We prove that in the case of projective sets it is possible to satisfy this for almost all codes using a continuous function. We also show that it is impossible to improve this to all codes even if more complex functions in the codes are allowed. We also study the intermediate steps of the Borel hierarchy, and we give an estimation for the complexity of such functions in the codes, which verify the Baire property for actually all codes.