Euclidean topology

Definition

The Euclidean norm on ${\displaystyle {\mathbb{R}}^n}$ is the non-negative function ${\displaystyle \Vert \cdot \Vert :{\mathbb{R}}^n\to \mathbb{R}}$ defined by $${\displaystyle \Vert \left(p_1,\dots ,p_n\right)\Vert :=\sqrt{p_1^2+\cdots +p_n^2}.}$$

Like all norms, it induces a canonical metric defined by ${\displaystyle d(p,q)=\Vert p-q\Vert .}$ 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 ${\displaystyle p=\left(p_1,\dots ,p_n\right)}$ and ${\displaystyle q=\left(q_1,\dots ,q_n\right)}$ is $${\displaystyle d(p,q)=\Vert p-q\Vert =\sqrt{{\left(p_1-q_1\right)}^2+{\left(p_2-q_2\right)}^2+\cdots +{\left(p_i-q_i\right)}^2+\cdots +{\left(p_n-q_n\right)}^2}.}$$

In any metric space, the open balls form a base for a topology on that space. 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.