Definition:Cartesian Product/Countable

Definition
Let $\left \langle {S_n} \right \rangle_{n \mathop \in \N}$ be an infinite sequence of sets.

The cartesian product of $\left \langle {S_n} \right \rangle$ is defined as:


 * $\displaystyle \prod_{k \mathop = 1}^\infty S_k = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S_k}\right\}$

It defines the concept:
 * $S_1 \times S_2 \times \cdots \times S_n \times \cdots$

Thus $\displaystyle \prod_{k \mathop = 1}^\infty S_k$ is the set of all infinite sequences $\left({x_1, x_2, \ldots, x_n, \ldots}\right)$ with $x_k \in S_k$.

Also see

 * Generalized Cartesian products of algebraic structures:
 * Definition:External Direct Product/General Definition
 * Definition:Internal Direct Product/General Definition
 * Definition:Group Direct Product/General Definition
 * Definition:Internal Group Direct Product/General Definition