Uniformization (set theory): Difference between revisions

In [[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

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 [[nonemptyEmpty 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.