Euclidean topology
Template:Short description In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on <math>n</math>-dimensional Euclidean space <math>\R^n</math> by the Euclidean metric.
DefinitionEdit
The Euclidean norm on <math>\R^n</math> is the non-negative function <math>\|\cdot\| : \R^n \to \R</math> defined by <math display=block>\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>d(p, q) = \|p - q\|.</math> The metric <math>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>p = \left(p_1, \ldots, p_n\right)</math> and <math>q = \left(q_1, \ldots, q_n\right)</math> is <math display=block>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.<ref>Metric space#Open and closed sets.2C topology and convergence</ref> The Euclidean topology on <math>\R^n</math> is the topology Template:Em by these balls. In other words, the open sets of the Euclidean topology on <math>\R^n</math> are given by (arbitrary) unions of the open balls <math>B_r(p)</math> defined as <math>B_r(p) := \left\{x \in \R^n : d(p,x) < r\right\},</math> for all real <math>r > 0</math> and all <math>p \in \R^n,</math> where <math>d</math> is the Euclidean metric.
PropertiesEdit
When endowed with this topology, the real line <math>\R</math> is a T5 space. Given two subsets say <math>A</math> and <math>B</math> of <math>\R</math> with <math>\overline{A} \cap B = A \cap \overline{B} = \varnothing,</math> where <math>\overline{A}</math> denotes the closure of <math>A,</math> there exist open sets <math>S_A</math> and <math>S_B</math> with <math>A \subseteq S_A</math> and <math>B \subseteq S_B</math> such that <math>S_A \cap S_B = \varnothing.</math><ref name="CEIT">Template:Citation</ref>