Definition[edit]

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


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


In any metric space, the open balls form a base for a topology on that space.^[1] The Euclidean topology on ${\displaystyle \mathbb {R} ^{n}}$ is the topology generated by these balls. In other words, the open sets of the Euclidean topology on ${\displaystyle \mathbb {R} ^{n}}$ are given by (arbitrary) unions of the open balls ${\displaystyle B_{r}(p)}$ defined as ${\displaystyle B_{r}(p):=\left\{x\in \mathbb {R} ^{n}:d(p,x)<r\right\},}$ for all real ${\displaystyle r>0}$ and all ${\displaystyle p\in \mathbb {R} ^{n},}$ where ${\displaystyle d}$ is the Euclidean metric.