# Euclidean topology

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 $S_{A}\cap S_{B}=\varnothing .$