Definition:Cartesian Product of Family/Definition 1

Definition
Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be a family of sets.

The Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ is the set of all families $\left\langle{s_i}\right\rangle_{i \mathop \in I}$ with $s_i \in S_i$ for each $i \in I$.

This can be denoted $\displaystyle \prod_{i \mathop \in I} S_i$ or, if $I$ is understood, $\displaystyle \prod_i S_i$.

Axiom of Choice
It is of the utmost importance that one be aware that for many $I$, establishing non-emptiness of $\displaystyle \prod_{i \mathop \in I} S_i$ requires a suitable version of the Axiom of Choice.

Details on this correspondence can be found on Equivalence of Versions of Axiom of Choice.

Also see

 * Equivalence of Definitions of Cartesian Product