# nLab Euclidean topology

### Context

#### Topology

topology

algebraic topology

# Contents

## Definition

###### Definition

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

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

• 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 $\mathbb{R}^k$ and $\mathbb{R}^l$ (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)