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

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

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]}

- Hilbert space – Type of topological vector space
- List of Banach spaces
- List of topologies – List of concrete topologies and topological spaces

**^**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