# Andrés Caicedo: Co-analytic uniformization

Wednesday, December 10 from 3 to 4pm
Room: Math 226
Speaker: Andrés Caicedo (BSU)
Title: Co-analytic uniformization

Abstract: It is an easy consequence of the axiom of choice that if X is an arbitrary set and R is a binary relation on X (a subset of $X^2$) then R admits a uniformization, that is, there is a function f whose domain is $\{x \in X : \text{there is a } y \in X \text{ with } x R y\}$ and such that for all x in its domain, x R f(x).

If X is the set of reals, and R is a reasonably definable relation, one might expect that the existence of such a function f can actually be established without using the axiom of choice.

We sketch a classical result independently due to Novikov and Kondo showing that this is indeed the case if R is Borel (and even if it is “slightly” more complicated than Borel).