# 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

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 then simply 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) , 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]