Definition:Euclidean Space/Euclidean Topology

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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 $M = \left({S^n, d}\right)$ be a Euclidean space.


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


Special Cases

Real Number Line

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.


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.


Real Vector Space

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

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


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


Rational Euclidean Space

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

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


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


Complex Euclidean Space

Let $\C$ be the complex plane.

Let $M = \left({\C, d}\right)$ be a complex Euclidean space.


The topology induced by the Euclidean metric $d$ 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, 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.