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.^{[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 and all , where is the Euclidean metric.

## Properties

- The real line, with this topology, is a T
_{5}space. Given two subsets, say*A*and*B*, of**R**with*A*∩*B*=*A*∩*B*= ∅, where*A*denotes the closure of*A*, there exist open sets*S*and_{A}*S*with_{B}*A*⊆*S*and_{A}*B*⊆*S*such that_{B}*S*∩_{A}*S*= ∅._{B}^{[2]}

## 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