Definition:Euclidean Space

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

Let $M = \left({\R^n, d}\right)$ where $\displaystyle d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$.

Then $M = \left({\R^n, d}\right)$ is a metric space.

Such a space is called a Euclidean $n$-space.

The metric $d$ is called the Euclidean Metric.

Any vector space for which a metric is defined that is precisely equivalent, for any two points, to the Euclidean metric is called a Euclidean space.

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

From Rational Numbers form Metric Space it follows from the above definition it follows that $\Q^n$ is also a Euclidean $n$-space.

Complex Euclidean Space
The set of complex numbers $\C$ is also a metric space, as is proved here.

Euclidean Topology
The topology induced by the Euclidean metric on a Euclidean space $M$ is called the Euclidean topology.

The Euclidean topology is sometimes called the usual topology.