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TheIn [[set theory]], a branch of [[mathematics]], the '''axiom of uniformization''', is a weak form of the [[axiom of choice]],. It states that if <math>R</math> is a [[subset]] of <math>X\times Y</math>, where <math>X</math> and <math>Y</math> are [[Polish space]]s, then there is a subset <math>f</math> of <math>R</math> that is a [[partial function]] from <math>X</math> to <math>Y</math>, and whose domain (the [[Set (mathematics)|set]] of all <math>x</math> such that <math>f(x)</math> exists) equals
: <math>\{x \in X| \mid \exists y \in Y: (x,y) \in R\}\,</math>▼
then there is a subset <math>f</math> of <math>R</math> that is a [[partial function]] from <math>X</math> to <math>Y</math>, and whose domain equals
▲: <math>\{x\in X|\exists y\in Y (x,y)\in R\}\,</math>
Such a function is called a '''uniformizing function''' for <math>R</math>, or a '''uniformization''' of <math>R</math>.
 
[[Image:Uniformization ill.png|thumb|right|Uniformization of relation ''R'' (light blue) by function ''f'' (red).]]
A [[pointclass]] <math>\boldsymbol{\Gamma}</math> is said to have the '''uniformization property''' if every relation <math>R</math> in <math>\boldsymbol{\Gamma}</math> can be uniformized by a partial function in <math>\boldsymbol{\Gamma}</math>. The uniformization property is implied by the [[scale property]], at least for [[adequate pointclass]]es.▼
 
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To see the relationship with the axiom of choice, observe that <math>R</math> can be thought of as associating, to each element of <math>X</math>, a subset of <math>Y</math>. A uniformization of <math>R</math> then picks exactly one element from each such subset, whenever the subset is [[Empty set|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]] <math>\boldsymbol{\Gamma}</math> is said to have the '''uniformization property''' if every [[Binary relation|relation]] <math>R</math> in <math>\boldsymbol{\Gamma}</math> can be uniformized by a partial function in <math>\boldsymbol{\Gamma}</math>. The uniformization property is implied by the [[scale property]], at least for [[adequate pointclass]]es of a certain form.
 
It follows from [[ZFC]] alone that <math>\boldsymbol{\Pi}^1_1</math> and <math>\boldsymbol{\Sigma}^1_2</math> have the uniformization property. It follows from the existence of sufficient [[large cardinal]]s that
*<math>\boldsymbol{\Pi}^1_{2n+1}</math> and <math>\boldsymbol{\Sigma}^1_{2n+2}</math> have the uniformization property for every [[natural number]] <math>n</math>.
*Therefore, the collection of [[projective set]]s 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.)
 
== References ==
 
* {{Book_reference | Author=Moschovakis, Yiannis N. | Title=Descriptive Set Theory | Publisher=North Holland | Year=1980 |ID=ISBN 0-444-70199-0}}
* {{cite book | author=Moschovakis, Yiannis N. |authorlink = Yiannis N. Moschovakis| title=Descriptive Set Theory | url=https://archive.org/details/descriptivesetth0000mosc | url-access=registration | publisher=North Holland | year=1980 |isbn=0-444-70199-0}}
 
[[Category:Set theory]]
[[Category:Descriptive set theory]]
[[Category:Axiom of choice]]
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