The space which satisfies all five of the

postulates
(aka

axioms) upon which

Euclid built his ideas of

geometry.

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 *S* 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).

The *N*-dimensional Euclidean space **E**^{N}
is the product space of *N* copies of **E**. For example, a set
in **E**^{2} is open if it is the union of open rectangles,
and a set in **E**^{3} is open if it is the union of open rectanglular
prisms.

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