Uniformization (set theory)

In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice.

are Polish spaces, then there is a subset

, and whose domain (the set of all

exists) equals Such a function is called a uniformizing function for

To see the relationship with the axiom of choice, observe that

can be thought of as associating, to each element 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.

is said to have the uniformization property if every relation

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 the existence of sufficient large cardinals that