# nLab Euclidean topology

### Context

#### Topology

topology

algebraic topology

# Contents

## Definition

###### Definition

For $n\in ℕ$ a natural number, write ${ℝ}^{n}$ for the Cartesian space of dimension $n$. The Euclidean topology is the topology on ${ℝ}^{n}$ characterized by the following equivalent statements

• it is the metric topology induced from the canonical structure of a metric space on ${ℝ}^{n}$ with distance function given by $d\left(x,y\right)=\sqrt{{\sum }_{i=1}^{n}\left({x}_{i}-{y}_{i}{\right)}^{2}}$;

• an open subset is precisely a subset such that contains an open ball around each of its points;

• it is the product topology induced from the standard topology on the real line.

## Properties

###### Proposition

Two Cartesian spaces ${ℝ}^{k}$ and ${ℝ}^{l}$ (with the Euclidean topology) are homeomorphic precisely if $k=l$.

A proof of this statement was an early success of algebraic topology.

Revised on January 15, 2011 04:56:57 by Toby Bartels (98.19.56.183)