page_title: Uniformization (set theory)
hierarchical_section_title: Uniformization (set theory)
caption_attribution_description: Made by myself with Xfig. Source included.
context_page_description: 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.
context_section_description: 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 (the set of all  such that  exists) 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 (rather than just Polish spaces) 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(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.)
caption: Uniformization of relation R (light blue) by function f (red).
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