Euclidean topology

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

DefinitionEdit

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.

PropertiesEdit

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 alsoEdit

ReferencesEdit

  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