Euclidean topology

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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.

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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 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

  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