In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on
n
{\displaystyle n}
-dimensional Euclidean space
n
{\displaystyle \mathbb {R} ^{n}}
by the Euclidean metric.
The Euclidean norm on
{\displaystyle \mathbb {R} ^{n}}
is the non-negative function
‖ ⋅ ‖ :
n
defined by
n
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
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
is the Euclidean metric.
When endowed with this topology, the real line
is a T5 space.
Given two subsets say
denotes the closure of
there exist open sets