Choice function

Definition

Let $X$ be a set of nonempty sets. A choice function on $X$ is a function $c \colon X \to \bigcup X$ such that $c(S) \in S$ for each $S \in X$.

Remarks

For a finite set $X$, a choice function exists without any axiom of choice. For the set of nonempty subsets of $\mathbb{N}$, the function that sends each subset to its least element is a choice function. The axiom of choice says that each set of nonempty sets has a choice function.

Explanations

L1 human

Think of $X$ as a collection of boxes, none empty. A choice function names one object in each box. Sometimes a rule gives it, such as "take the smallest number".