Euclidean topology

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

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

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

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

Properties
When endowed with this topology, the real line $$\R$$ is a T5 space. Given two subsets say $$A$$ and $$B$$ of $$\R$$ with $$\overline{A} \cap B = A \cap \overline{B} = \varnothing,$$ where $$\overline{A}$$ 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 $$S_A \cap S_B = \varnothing.$$