Product set/Arbitrary/Definition

Let ${\displaystyle {}I}$ be a set, and let, for every ${\displaystyle {}i\in I}$, a set ${\displaystyle {}M_{i}}$ be given. Then the set

${\displaystyle {}M=\prod _{i\in I}M_{i}=\{(x_{i})_{i\in I}:\,x_{i}\in M_{i}{\text{ for all }}i\in I\}\,}$

is called the product set of the ${\displaystyle {}M_{i}}$.