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 is a subset of, where and are Polish spaces, then there is a subset of that is a partial function from to, and whose domain equals
Such a function is called a uniformizing function for, or a uniformization of.
To see the relationship with the axiom of choice, observe that can be thought of as associating, to each element of, a subset of. A uniformization of then picks exactly one element from each such subset, whenever the subset is non-empty. Thus, allowing arbitrary sets X and Y would make the axiom of uniformization equivalent to the axiom of choice.
A pointclass is said to have the uniformization property if every relation 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
- and have the uniformization property for every natural number.
- Therefore, the collection of projective sets has the uniformization property.
- Every relation in L can be uniformized, but not necessarily by a function in L. In fact, L does not have the uniformization property.
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