nLab
Euclidean topology

Contents

Definition

Definition

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

Properties

Proposition

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