\begin{definition}[Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable]
The countable cartesian product defined as:
:$\ds \R^\omega := \R \times \R \times \cdots = \prod_\N \R$
is called the '''countable-dimensional real cartesian space'''.
Thus, $\R^\omega$ can be defined as the set of all real sequences:
:$\R^\omega = \set {\sequence {x_1, x_2, \ldots}: x_1, x_2, \ldots \in \R}$
\end{definition}