\begin{definition}[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 $\ds \prod_{i \mathop \in I} S_i$ requires a suitable version of the Axiom of Choice.
Details on this correspondence can be found on Equivalence of Versions of Axiom of Choice.
{{explain|What is meant by "for many $I$"? Clarification needed: which $I$ require which versions? The link cited does not help.}}
Category:Definitions/Cartesian Product
Category:Definitions/Indexed Families
\end{definition}