Definition:Metrizable Topology/Definition 2

Definition

Let $T = \struct {S, \tau}$ be a topological space.

$T$ is said to be metrizable if and only if there exists a metric space $M = \struct{A, d}$ such that:

$T$ is homeomorphic to the topological space $\struct{A, \tau_d}$

where $\tau_d$ is the topology induced by $d$ on $A$.

Also see

• Definition:Metrizable Topology/Definition 1

• Equivalence of Definitions of Metrizable Topology

• Indiscrete Topology is not Metrizable: thus, not all topological spaces are metrizable

• Definition:Completely Metrizable Topology

• Results about metrizable topologies can be found here.

Linguistic Note

The UK English spelling of metrizable is metrisable, but it is rarely found.

Sources

This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code.

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces