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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.

Source of Name

This entry was named for René Descartes.

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products