\begin{definition}[Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2]
Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be an 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:
:$\forall i \in I: S_i = S$
The '''Cartesian space of $S$ indexed by $I$''' is defined and denoted as:
:$\ds S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$
\end{definition}