Definition:Cartesian Product/Cartesian Space/Family of Sets

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:
 * $\forall i \in I: S_i = S$

Then $\displaystyle \prod_{i \mathop \in I} S_i$ can be denoted $S^I$ and is the Cartesian space of $S$ indexed by $I$.

Also denoted as
It is reported that some sources give this as the Cartesian $I$-space of $S$.

On it is generally referred to as an indexed Cartesian space.