Euclidean topology
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. 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

Topology
Euclid