Uniformization (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 $Y$ 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 $R$.

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.