nLab
metrisable topological space

Contents

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

A topological space (X,τ)(X,\tau) is called metrisable if there exists the stucture of a metric space (X,d)(X,d) on the underlying set, such that τ\tau is the corresponding metric topology.

Properties

Metrizable spaces enjoy a number of separation properties: they are Hausdorff, regular, and even normal. They are also paracompact.

Metrizable spaces are closed under topological coproducts and of course subspaces (and therefore equalizers); they are closed under countable products but not general products (for instance, a product of uncountably many copies of the real line \mathbb{R} is not a normal space).

Metrisability theorem

Fundamental early work in point-set topology established a number of metrization theorems, i.e., theorems which give sufficient conditions for a space to be metrisable. One of the more useful theorems is Urysohn metrization theorem: A regular, Hausdorff, and second-countable space is metrisable. So, for instance, a compact Hausdorff space that is second-countable is metrisable. Other metrization theorems are:

  • Nagata-Smirnov metrization theorem

  • Bing metrization theorem

  • Moore metrization theorem

References

See also

Last revised on March 15, 2019 at 17:27:27. See the history of this page for a list of all contributions to it.