Euclidean topology

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

Definition

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.

Properties

When endowed with this topology, the real line ${\displaystyle \mathbb {R} }$ is a T₅ space. Given two subsets say ${\displaystyle A}$ and ${\displaystyle B}$ of ${\displaystyle \mathbb {R} }$ with ${\displaystyle {\overline {A}}\cap B=A\cap {\overline {B}}=\varnothing ,}$ where ${\displaystyle {\overline {A}}}$ denotes the closure of ${\displaystyle A,}$ there exist open sets ${\displaystyle S_{A}}$ and ${\displaystyle S_{B}}$ with ${\displaystyle A\subseteq S_{A}}$ and ${\displaystyle B\subseteq S_{B}}$ such that ${\displaystyle S_{A}\cap S_{B}=\varnothing .}$^[2]

See also

• Hilbert space – Type of topological vector space
• List of Banach spaces
• List of topologies – List of concrete topologies and topological spaces

References