Uniformization (set theory)

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if $R$ is a subset of $X\times Y$, where $X$ and $Y$ are Polish spaces, then there is a subset $f$ of $R$ that is a partial function from $X$ to $Y$, and whose domain equals

$\{x\in X|\exists y\in Y(x,y)\in R\}\,$

Such a function is called a uniformizing function for $R$, or a uniformization of $R$.

File:Unif small.jpg

Uniformization of relation R by function f

To see the relationship with the axiom of choice, observe that $R$ can be thought of as associating, to each element of $X$, a subset of $Y$. A uniformization of $R$ then picks exactly one element from each such subset, whenever the subset is nonempty.

A pointclass ${\boldsymbol {\Gamma }}$ is said to have the uniformization property if every relation $R$ in ${\boldsymbol {\Gamma }}$ can be uniformized by a partial function in ${\boldsymbol {\Gamma }}$. The uniformization property is implied by the scale property, at least for adequate pointclasses.

References

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