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.

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.