Definition:Euclidean Space

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

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

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.