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