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 defined by Like all norms, it induces a canonical metric defined by The metric induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points and is 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 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 where denotes the closure of A, there exist open sets S_A and S_B with and such that