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

Definition

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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.

Properties

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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  [2]

See also

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References

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  1. ^ Metric space#Open and closed sets.2C topology and convergence
  2. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X