nLab
Euclidean topology

Context

Topology

topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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)