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 ${\displaystyle B_{r}(p):=\left\{x\in \mathbb {R} ^{n}:d(p,x)<r\right\}}$, 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[]

• Euclidean norm
• List of topologies – List of concrete topologies and topological spaces

References[]