Topological structure of Euclidean space
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on
-dimensional Euclidean space
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, the open balls form a base for a topology on that space.[1] The Euclidean topology on
is the topology generated 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
[2]
See also
- Hilbert space – Type of topological vector space
- List of Banach spaces
- List of topologies – List of concrete topologies and topological spaces
References
- ^ Metric space#Open and closed sets.2C topology and convergence
- ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X