Definition:Euclidean Space/Euclidean Topology

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 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
Bear in mind that Euclid himself did not in fact conceive of the Euclidean space as defined here. It is called that because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.