Euclidean topology explained

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 \| ⋅ \| : \R ^{n} \to \R defined by$$\backslash left\backslash |\backslash left(p\_1,\; \backslash ldots,\; p\_n\backslash right)\backslash right\backslash |\; ~:=~\; \backslash sqrt.$$

Like all norms, it induces a canonical metric defined by

d(p, q) = \|p - q\|. The metric d : \R ^{n} x \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 $$d(p,\; q)\; ~=~\; \backslash |p\; -\; q\backslash |\; ~=~\; \backslash sqrt.$$

In any metric space, the open balls form a base for a topology on that space.^[1] 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.

Notes and References

1. [Metric space|Metric space#Open and closed sets.2C topology and convergence]