\begin{definition}[Definition:Cartesian Product/Countable]
Let $\sequence {S_n}_{n \mathop \in \N}$ be an infinite sequence of sets. 
The '''cartesian product''' of $\sequence {S_n}$ is defined as:
:$\ds \prod_{k \mathop = 1}^\infty S_k = \set {\tuple {x_1, x_2, \ldots, x_n, \ldots}: \forall k \in \N: x_k \in S_k}$
It defines the concept:
:$S_1 \times S_2 \times \cdots \times S_n \times \cdots$
Thus $\ds \prod_{k \mathop = 1}^\infty S_k$ is the set of all infinite sequences $\tuple {x_1, x_2, \ldots, x_n, \ldots}$ with $x_k \in S_k$.
\end{definition}