Definition:Cartesian Product/Family of Sets

Axiom of Choice
It is of the utmost importance that one be aware that for many $I$, establishing non-emptiness of $\displaystyle \prod_{i \mathop \in I} S_i$ requires a suitable version of the Axiom of Choice.

Details on this correspondence can be found on Equivalence of Versions of Axiom of Choice.

Also see

 * Equivalence of Versions of Axiom of Choice