Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable

Definition
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:


 * $\R^\omega = \set {\sequence {x_1, x_2, \ldots}: x_1, x_2, \ldots \in \R}$

Also known as
The countable-dimensional real cartesian space can be given the more precise name countably-infinite-dimensional real cartesian space, but this is generally unnecessarily unwieldy.

Some sources call this (countably) infinite-dimensional Euclidean $n$-space or countable real Euclidean space -- however, on this term is reserved for the associated metric space.

Beware that some sources omit the qualifier countable or countably, thereby leaving the opportunity for confusing with the uncountable version of this space.