Euclidean spaces are nowadays defined in terms of Cartesian coordinates, but they have various definitions.

**E** as a topological space

The one-dimensional Euclidean space **E** is defined on the set of
real numbers taken in their natural order.

Given some real numbers r, e, consider the set ** {n| r-e < n
< r+e}**.

Let us construct the collection

**of all such subsets for any r and any e. The Euclidean topology is the topology for which**

*S***is a base. (That is, any open interval or any combination of open intervals is considered "open" in this topology).**

*S*The *N*-dimensional Euclidean space **E^{N}**
is the product space of

*N*copies of

**E**. For example, a set in

**E**is open if it is the union of open rectangles, and a set in

^{2}**E**is open if it is the union of open rectanglular prisms.

^{3}Several metric spaces induce the topology **E^{N}**.
(examples here)