Set theory | Descriptive set theory
Uniformization (set theory)
In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if is a subset of , where and are Polish spaces, then there is a subset of that is a partial function from to , and whose domain (the set of all such that exists) equals Such a function is called a uniformizing function for , or a uniformization of . 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.) (Wikipedia).
