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

## Definition

Let $\R$ denote the real number line.

Let $d: \R \times \R \to \R$ denote the Euclidean metric on $\R$.

Let $\tau_d$ denote the topology on $\R$ induced by $d$.

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

Hence $\struct {\R, \tau_d}$ is referred to as the **real number line with the Euclidean topology**.

## Also known as

The **Euclidean topology**, when applied to a real Cartesian space, is often referred to as the **usual topology**.

## Also see

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

- Results about
**the real number line with the Euclidean topology**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, Euclidean topology 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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 3.2$: Topological Spaces: Example $1$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology