Euclidean topology

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In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space Rn 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 Rn is then simply the topology generated by these balls. In other words, the open sets of the Euclidean topology on Rn are given by arbitrary union of the open balls , for all , where d is the Euclidean metric.


  • 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]