Uniformization (set theory)

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R {displaystyle R} is a subset of X × Y {displaystyle X imes 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 (in the sense of the set of all x {displaystyle x} such that f ( x ) {displaystyle f(x)} exists) equals In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R {displaystyle R} is a subset of X × Y {displaystyle X imes 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 (in the sense of the set of all x {displaystyle x} such that f ( x ) {displaystyle f(x)} 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 nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC. A pointclass Γ {displaystyle {oldsymbol {Gamma }}} is said to have the uniformization property if every relation R {displaystyle R} in Γ {displaystyle {oldsymbol {Gamma }}} can be uniformized by a partial function in Γ {displaystyle {oldsymbol {Gamma }}} . The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form. It follows from ZFC alone that Π 1 1 {displaystyle {oldsymbol {Pi }}_{1}^{1}} and Σ 2 1 {displaystyle {oldsymbol {Sigma }}_{2}^{1}} have the uniformization property. It follows from the existence of sufficient large cardinals that