# Definition:Euclidean Space/Euclidean Topology/Real

## Contents

## Definition

Let $\R^n$ be an $n$-dimensional real vector space.

Let $M = \left({\R^n, d}\right)$ be a real Euclidean $n$-space.

The topology $\tau_d$ induced by the Euclidean metric $d$ is called the **Euclidean topology**.

### Real Number Line

Let $\R$ be the set of real numbers.

Let $M = \left({\R, d}\right)$ be the real number line under the Euclidean metric $d$.

The topology $\tau_d$ induced by $d$ is called the **Euclidean topology**.

### Real Number Plane

Let $\R^n$ be an $n$-dimensional real vector space.

Let $M = \left({\R^2, d}\right)$ be a real Euclidean space of $2$ dimensions.

The topology $\tau_d$ induced by the Euclidean metric $d$ is called the **Euclidean topology**.

The space $\left({\R^2, \tau_d}\right)$ is known as the **(real) Euclidean plane**.

## Also known as

The **Euclidean topology** is often called the **usual topology**.

## Also see

- Results about
**Euclidean spaces**can be found here.

## Source of Name

This entry was named for Euclid.

## Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean Space and Euclidean norm.

They bear that name because the geometric space which it gives rise to is **Euclidean** in the sense that it is consistent with Euclid's fifth postulate.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 28: \ 9$