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 $$R$$ 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
 * $$\{x \in X \mid \exists y \in Y: (x,y) \in R\}\,$$

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 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 $$\boldsymbol{\Gamma}$$ is said to have the uniformization property if every relation $$R$$ in $$\boldsymbol{\Gamma}$$ can be uniformized by a partial function in $$\boldsymbol{\Gamma}$$. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.

It follows from ZFC alone that $$\boldsymbol{\Pi}^1_1$$ and $$\boldsymbol{\Sigma}^1_2$$ have the uniformization property. It follows from the existence of sufficient large cardinals that
 * $$\boldsymbol{\Pi}^1_{2n+1}$$ and $$\boldsymbol{\Sigma}^1_{2n+2}$$ 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 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.)