In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R is a subset of X × 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 (in the sense of the set of all x such that f ( x ) exists) equals
{ x ∈ X | ∃ y ∈ Y ( x , y ) ∈ R } 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 nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.
A pointclass Γ is said to have the uniformization property if every relation R in Γ can be uniformized by a partial function in Γ . The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.
It follows from ZFC alone that Π 1 1 and Σ 2 1 have the uniformization property. It follows from the existence of sufficient large cardinals that
Π 2 n + 1 1 and Σ 2 n + 2 1 have the uniformization property for every natural number n .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 AC. The point is that every such relation can be uniformized in some transitive inner model of V in which AD holds.)
