Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1

Definition
Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an family of sets indexed by $I$.

Let $\displaystyle \prod_{i \mathop \in I} S_i$ 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 the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S$ for each $i \in I$:
 * $S_I := \displaystyle \prod_I S = \set {\family {s_i}_{i \mathop \in I}: s_i \in S}$