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In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if R {\displaystyle R} is a subset of X × Y {\displaystyle X\times Y} , where X {\displaystyle X} and Y {\displaystyle Y} are Polish spaces, then there is a subset f {\displaystyle f} of R {\displaystyle R} that is a partial function from X {\displaystyle X} to Y {\displaystyle Y} , and whose domain {\displaystyle f} exists] equals
Such a function is called a uniformizing function for R {\displaystyle R} , or a uniformization of R {\displaystyle R}.
To see the relationship with the axiom of choice, observe that R {\displaystyle R} can be thought of as associating, to each element of X {\displaystyle X} , a subset of Y {\displaystyle Y}. A uniformization of R {\displaystyle R} 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 Γ {\displaystyle {\boldsymbol {\Gamma }}} is said to have the uniformization property if every relation R {\displaystyle R} in Γ {\displaystyle {\boldsymbol {\Gamma }}} can be uniformized by a partial function in Γ {\displaystyle {\boldsymbol {\Gamma }}}. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.