Such a function is called a uniformizing function for , or a uniformization of .
Uniformization of relation R (light blue) by function f (red).
To see the relationship with the axiom of choice, observe that can be thought of as associating, to each element of , a subset of . A uniformization of then picks exactly one element from each such subset, whenever the subset is
non-empty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice.
A
pointclass is said to have the uniformization property if every
relation in can be uniformized by a partial function in . The uniformization property is implied by the
scale property, at least for
adequate pointclasses of a certain form.
It follows from
ZFC alone that and have the uniformization property. It follows from the existence of sufficient
large cardinals that
and have the uniformization property for every
natural number.
Therefore, the collection of
projective sets has the uniformization property.
Every relation in
L(R) can be uniformized, but not necessarily by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).
(Note: it's trivial that every relation in L(R) can be uniformized in V, assuming V satisfies the axiom of choice. The point is that every such relation can be uniformized in some transitive inner model of V in which the
axiom of determinacy holds.)