Definition: Axiom of Choice. Letbe a collection of nonempty sets. Then we can choose a element from each setso that there exists a function(called a choice function) defined onwith the property that, for each set

Initially controversial, it is now a basic assumption used in many areas of maths. It is independent of the other axioms of set theory. Thus there are no contradictions in choosing to reject it and choosing another instead. The axiom can be stated in many equivalent ways. For example:

Examples of choice functions