Uniformization (set theory)

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if $R$ is a subset of $X\; imes\; Y$, 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: $\{xin\; X|exists\; yin\; Y\; (x,y)in\; R\},$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 nonempty. Thus, allowing arbitrary sets "X" and "Y" (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.

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.

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 $n$.
*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.)

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