Uniformity results on the Baire property

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.