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, such that

$\forall x\in X\exists y\in Y(x,y)\in R$

then there is a subset $f$ of $R$ that is a total function from $X$ to $Y$. Such a function is called a uniformizing function for $R$, or a uniformization of $R$.