# Definition:Euclidean Space

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## Definition

Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$.

Let $d: S^n \times S^n \to \R$ be the usual (Euclidean) metric on $S^n$.

Then $\tuple {S^n, d}$ is a **Euclidean space**.

## Special Cases

### Real Vector Space

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

Let the Euclidean metric $d$ be applied to $\R^n$.

Then $\struct {\R^n, d}$ is a **(real) Euclidean $n$-space**.

### Rational Euclidean Space

Let $\Q^n$ be an $n$-dimensional vector space of rational numbers.

Let the Euclidean Metric $d$ be applied to $\Q^n$.

Then $\struct {\Q^n, d}$ is a **(rational) Euclidean $n$-space**.

### Complex Euclidean Space

Let $\C$ be the complex plane.

Let $d$ be the Euclidean metric on $\C$.

Then $\struct {\C, d}$ is a **(complex) Euclidean space**.

## Euclidean Topology

Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$.

Let $M = \struct {S^n, d}$ be a Euclidean space.

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

## Euclidean Plane

For any real number $a$ let:

- $L_a = \set {\tuple {x, y} \in \R^2: x = a}$

Furthermore, define:

- $L_A = \set {L_a: a \in \R}$

For any two real numbers $m$ and $b$ let:

- $L_{m, b} = \set {\tuple {x, y} \in \R^2: y = m x + b}$

Furthermore, define:

- $L_{M, B} = \set {L_{m, b}: m, b \in \R}$

Finally let:

- $L_E = L_A \cup L_{M, B}$

The abstract geometry $\struct {\R^2, L_E}$ is called the **Euclidean plane**.

## 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, 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

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets (in passing, as an example) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Euclidean space** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Euclidean space** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Euclidean space (Cartesian space)**