# Euclidean topology

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In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$ by the Euclidean metric.

## Definition

In any metric space, the open balls form a base for a topology on that space.  The Euclidean topology on $\mathbb {R} ^{n}$ is then simply the topology generated by these balls. In other words, the open sets of the Euclidean topology on $\mathbb {R} ^{n}$ are given by (arbitrary) unions of the open balls $B_{r}(p)$ defined as $B_{r}(p):=\left\{x\in \mathbb {R} ^{n}:d(p,x) , for all real $r>0$ and all $p\in \mathbb {R} ^{n},$ where $d$ is the Euclidean metric.

## Properties

When endowed with this topology, the real line $\mathbb {R}$ is a T5 space. Given two subsets say $A$ and $B$ of $\mathbb {R}$ with ${\overline {A}}\cap B=A\cap {\overline {B}}=\varnothing ,$ where ${\overline {A}}$ denotes the closure of $A,$ there exist open sets $S_{A}$ and $S_{B}$ with $A\subseteq S_{A}$ and $B\subseteq S_{B}$ such that $S_{A}\cap S_{B}=\varnothing .$ ## Related Research Articles

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1. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology , Dover, ISBN   0-486-68735-X