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In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space \mathbb^n by the Euclidean metric. In any metric space, the open balls form a base for a topology on that space.Metric space#Open and closed sets.2C topology and convergence The Euclidean topology on \mathbb^n is then simply the topology ''generated'' by these balls. In other words, the open sets of the Euclidean topology on \mathbb^n are given by (arbitrary) unions of the open balls B_(p) defined as B_(p) := \, for all r > 0 and all p \in \mathbb^n, where d is the Euclidean metric.


Properties


* The real line, with this topology, is a T5 space. Given two subsets, say ''A'' and ''B'', of R with , where ''A'' denotes the closure of ''A'', there exist open sets ''SA'' and ''SB'' with and such that


References


{{Reflist Category:Topology