\begin{definition}[Definition:Cartesian Product/Cartesian Space/Family of Sets]
Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
Let $\ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
Let $S$ be a set such that:
:$\forall i \in I: S_i = S$
\end{definition}