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

The Euclidean norm on $\mathbb {R} ^{n}$ is the non-negative function $\|\cdot \|:\mathbb {R} ^{n}\to \mathbb {R}$ defined by $\left\|\left(p_{1},\ldots ,p_{n}\right)\right\|~:=~{\sqrt {p_{1}^{2}+\cdots +p_{n}^{2}}}.$

Like all norms, it induces a canonical metric defined by $d(p,q)=\|p-q\|.$ The metric $d:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R}$ induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points $p=\left(p_{1},\ldots ,p_{n}\right)$ and $q=\left(q_{1},\ldots ,q_{n}\right)$ is $d(p,q)~=~\|p-q\|~=~{\sqrt {\left(p_{1}-q_{1}\right)^{2}+\left(p_{2}-q_{2}\right)^{2}+\cdots +\left(p_{i}-q_{i}\right)^{2}+\cdots +\left(p_{n}-q_{n}\right)^{2}}}.$

In any metric space, the open balls form a base for a topology on that space.^[1] The Euclidean topology on $\mathbb {R} ^{n}$ is 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 $B_{r}(p):=\left\{x\in \mathbb {R} ^{n}:d(p,x)<r\right\},$ 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 $S_{A}\cap S_{B}=\varnothing .$ ^[2]