In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space $\backslash mathbb^n$ 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 $\backslash mathbb^n$ is then simply the topology ''generated'' by these balls. In other words, the open sets of the Euclidean topology on $\backslash mathbb^n$ are given by (arbitrary) unions of the open balls $B\_(p)$ defined as $B\_(p)\; :=\; \backslash $, for all $r\; >\; 0$ and all $p\; \backslash in\; \backslash mathbb^n$, where $d$ is the Euclidean metric.

** Properties **

* The real line, with this topology, is a T_{5} space. Given two subsets, say ''A'' and ''B'', of R with , where ''A'' denotes the closure of ''A'', there exist open sets ''S_{A}'' and ''S_{B}'' with and such that

** References **

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Category:Topology