Let \((X_\alpha)_{\alpha\in A}\) be a nonempty family of nonempty sets. Then there exists a function \(c\colon A\to \bigcup_{\alpha\in A} X_\alpha\) such that \(c(\alpha)\in X_\alpha\).
The function \(c\) is often called choice function.
Remarks
- The axiom of choice is equivalent to Zorns lemma .
- The axiom of choice is equivalent to the well-ordering theorem .
- The axiom of choice ensures the nonemptyness of Cartesian products .