Definition:Cartesian Product/Family of Sets

Uncountable Cartesian Product
Using this notation, it is then possible to define the Cartesian product of an uncountable family:

Also denoted as
Some sources use $\def \bigtimes {\mathop {\vcenter {\huge \times} } } \bigtimes \limits_{i \mathop \in I} S_i$ for $\ds \prod_{i \mathop \in I} S_i$.

If $S_i=S$ for all $i\in I$, then we also write $S^I := \prod_{i \mathop \in I} S_i$.

Also see

 * Definition:Indexed Cartesian Space: $\ds \prod_{i \mathop \in I} S_i$ where $\forall i \in I: S_i = S$, denoted $S^I$


 * Equivalence of Definitions of Cartesian Product of Indexed Family


 * Equivalence of Formulations of Axiom of Choice