# Definition:Euclidean Space/Euclidean Topology/Real Number Line

< Definition:Euclidean Space | Euclidean Topology(Redirected from Definition:Euclidean Topology on Real Number Line)

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## Definition

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**.

## Also known as

The **Euclidean topology** is sometimes 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): $\text{II}: \ 28$