# Definition:Cartesian Product/Cartesian Space/Real Cartesian Space

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## Contents

## Definition

Let $n \in \N_{>0}$.

Then $\R^n$ is the cartesian product defined as follows:

- $\displaystyle \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$

Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:

- $\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$

### Countable-Dimensional Real Cartesian Space

The countable cartesian product defined as:

- $\displaystyle \R^\omega := \R \times \R \times \cdots = \prod_\N \R$

is called the **countable-dimensional real cartesian space**.

Thus, $\R^\omega$ can be defined as the set of all real sequences:

- $\R^\omega = \left\{{\left({x_1, x_2, \ldots}\right): x_1, x_2, \ldots \in \R}\right\}$

## Also known as

The **real cartesian space** of order $n$ is sometimes seen as the **(real) cartesian $n$-space**.

Some sources call this **euclidean $n$-space** -- however, on $\mathsf{Pr} \infty \mathsf{fWiki}$ this term is reserved for the associated metric space.

## Also see

It can be shown that:

- $\R^2$ is isomorphic to any infinite flat plane in space
- $\R^3$ is isomorphic to the whole of space itself.

## Source of Name

This entry was named for René Descartes.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.4$