Euclidean topology

Short description: Topological structure of Euclidean space

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

Definition

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

Like all norms, it induces a canonical metric defined by $d(p,q)=\Vert p-q\Vert .$ The metric $d:{\mathbb{ℝ}}^n\times {\mathbb{ℝ}}^n\to \mathbb{ℝ}$ induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points $p=(p_1,\dots ,p_n)$ and $q=(q_1,\dots ,q_n)$ is $$d(p,q)=\Vert p-q\Vert =\sqrt{{(p_1-q_1)}^2+{(p_2-q_2)}^2+\cdots +{(p_i-q_i)}^2+\cdots +{(p_n-q_n)}^2}.$$

In any metric space, the open balls form a base for a topology on that space. The Euclidean topology on ${\mathbb{ℝ}}^n$ is the topology generated by these balls. In other words, the open sets of the Euclidean topology on ${\mathbb{ℝ}}^n$ are given by (arbitrary) unions of the open balls $B_r(p)$ defined as $B_r(p):=\{x\in {\mathbb{ℝ}}^n:d(p,x)<r\},$ for all real $r>0$ and all $p\in {\mathbb{ℝ}}^n,$ where $d$ is the Euclidean metric.

Properties

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

See also

• Hilbert space – Generalization of Euclidean space allowing infinite dimensions
• List of Banach spaces
• List of topologies – List of concrete topologies and topological spaces