In mathematics, and especially general topology
, the Euclidean topology is the natural topology
induced on Euclidean n-space
by the Euclidean metric
In any metric space
, 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 then simply 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
, for all
is the Euclidean metric.
* The real line, with this topology, is a T5 space
. Given two subsets, say ''A'' and ''B'', of R with , where ''A''
denotes the closure
of ''A'', there exist open sets ''SA
'' and ''SB
'' with and such that