# Euclidean topology

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*= ∅.[2]_{B}

## 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

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