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.)