direct product

If two sets $A$  and $B$ are given, we are able to make ordered combinations of each element of the set. Namely if $a\in A$  and $b\in B$ , one of ordered combinations is $(a,b)$ . These are called direct product. We will show all combinations $A\times  B$ . Hence,

$A\times B=\left\{ (a,b) | a\in A, b\in B \right\}$

You have to note that direct product is ordered. Therefore, $(a,b)$  is not equal to $(b,a)$ . Because two sets are arbitrary, $A=B$  is allowed. If $A=B=\mathbb{R}$ , $(a,b)$ means the point of Cartesian coordinates of $\mathbb{R}\times \mathbb{R}=\mathbb{R}^2$ .  Then, elements $a$  and $b$  become coordinate axes.

Direct product is expanded over two sets. Given n sets, it means n-dimensional space. You must have known $\mathbb{R}^3$ very well. Having already proved, you may still remember that, for $C=\left\{ 0,1 \right\}$   ,
$C^{\infty}=\left\{ 0,1 \right\}\times \left\{ 0,1 \right\}\times \cdots$
is uncountable.