In mathematics, and especially general topology, the **Euclidean topology** is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.

## Definition

In any metric space, the open balls form a base for a topology on that space.^{[1]}
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 defined as , for all real and all where is the Euclidean metric.

## Properties

When endowed with this topology, the real line is a T_{5} 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

## 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