nLab
metric topology

Context

Topology

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

see also differential topology, 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

Given a metric space (X,d)(X,d), the metric topology on XX is the structure of a topological space on XX which is generated from the basis of a topology given by the open balls

B(x,r){xX|d(x,y)<r} B(x,r) \coloneqq \{x \in X \;|\; d(x,y) \lt r \}

for all xXx \in X and r(0,)r \in (0,\infty) \subset \mathbb{R}.

A topological space whose topology is the metric topology for some metric space structure on its underlying set is called a metrizable topological space.

Revised on June 21, 2017 04:25:17 by Urs Schreiber (131.220.184.222)