David Schrittesser (Vienna)

Title: Ideals with and without definable selectors
 

Abstract:

Let X be a Polish space, and consider an ideal I on X. We can form the quotient of the Borel subsets of X by the equivalence relation which identifies sets which agree up to a set in I. For which ideals is it possible to select, in some definable manner, a representative for each equivalence class? One example where this is possible is the ideal of Lebesgue null sets: This follows from Lebesgue's Density Theorem. Another example is the meager ideal. In this talk I will present an abstract property of ideals which ensures that a selector can be defined; on the other hand, no definable selector exists for the ideal of countable sets. I will further present some consistency results.
(Joint work with Sandra Müller, Philipp Schlicht, and Thilo Weinert.)