In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if R is a subset of X\times Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X to Y, and whose domain (the set of all x such that f(x) exists) equals Such a function is called a uniformizing function for R, or a uniformization of R. To see the relationship with the axiom of choice, observe that R can be thought of as associating, to each element of X, a subset of Y. A uniformization of R 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 R 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