Euclidean topology

Short description: Topological structure of Euclidean space

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

Definition

The Euclidean norm on [math]\displaystyle{ \R^n }[/math] is the non-negative function [math]\displaystyle{ \|\cdot\| : \R^n \to \R }[/math] defined by [math]\displaystyle{ \left\|\left(p_1, \ldots, p_n\right)\right\| ~:=~ \sqrt{p_1^2 + \cdots + p_n^2}. }[/math]

Like all norms, it induces a canonical metric defined by [math]\displaystyle{ d(p, q) = \|p - q\|. }[/math] The metric [math]\displaystyle{ d : \R^n \times \R^n \to \R }[/math] induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points [math]\displaystyle{ p = \left(p_1, \ldots, p_n\right) }[/math] and [math]\displaystyle{ q = \left(q_1, \ldots, q_n\right) }[/math] is [math]\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}. }[/math]

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

Properties

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

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