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In mathematics, and especiallygeneral topology, theEuclidean topology is thenatural topology induced on${\displaystyle n}$-dimensionalEuclidean space${\displaystyle {\mathbb{R}}^n}$ by theEuclidean metric.

TheEuclidean 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 allnorms, it induces a canonicalmetric 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 theEuclidean norm is called theEuclidean metric or theEuclidean 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 anymetric space, theopen balls form abase for a topology on that space.^[1] The Euclidean topology on${\displaystyle {\mathbb{R}}^n}$ is the topologygenerated 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 for all real${\displaystyle r>0}$ and all${\displaystyle p\in {\mathbb{R}}^n,}$ where${\displaystyle d}$ is the Euclidean metric.

When endowed with this topology, the real line${\displaystyle \mathbb{R}}$ is aT₅ 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 theclosure 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]

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

• Topology
• Euclid

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