Let \((X_\alpha)_{\alpha\in A}\) be a family of sets. The Cartesian product is the set of all functions \(x\colon A\to \bigcup_{\alpha\in A} X_\alpha\) such that \(x_\alpha\in X_\alpha\). We denote the Cartesian product with \(\prod_{\alpha\in A} X_\alpha\). It is equipped with canonical projection maps \(\pi_\beta\colon \prod_{\alpha\in A} X_\alpha \to X_\beta\) defined by \(\phi_\beta(x)=x_\beta\).
Remarks
- According to the axiom of choice the Cartesian product is nonempty provided that \(A\) and all \(X_\alpha\) are nonempty.