In mathematics, and especially
general topology, the Euclidean topology is the
natural topology induced on
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
by the
Euclidean metric.
Definition
The
Euclidean norm on
is the non-negative function
defined by
Like all
norms, it induces a canonical
metric defined by
The metric
induced by the
Euclidean norm is called the
Euclidean metric or the
Euclidean distance and the distance between points
and
is
In any
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, the
open balls form a
base for a topology on that space.
[ Metric space#Open and closed sets.2C topology and convergence]
The Euclidean topology on
is the topology by these balls.
In other words, the open sets of the Euclidean topology on
are given by (arbitrary) unions of the open balls
defined as
for all real
and all
where
is the Euclidean metric.
Properties
When endowed with this topology, the real line
is a
T5 space.
Given two subsets say
and
of
with
where
denotes the
closure of
there exist open sets
and
with
and
such that
See also
*
*
*
References
{{reflist
Topology
Euclid