Definition:Cartesian Product/Cartesian Space/Family of Sets

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Let $I$ be an indexing set.

Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be an family of sets indexed by $I$.

Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$.

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 $I$-space of $S$.