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 EN
is the product space of N copies of E. For example, a set
in E2 is open if it is the union of open rectangles,
and a set in E3 is open if it is the union of open rectanglular
prisms.
Several metric spaces induce the topology EN.
(examples here)