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.

PropertiesEdit

  • The real line, with this topology, is a T5 space. Given two subsets, say A and B, of R with AB = AB = ∅, where A denotes the closure of A, there exist open sets SA and SB with ASA and BSB such that SASB = ∅.[2]

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