Definition:Metrizable Topology/Definition 1

Definition

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

$T$ is said to be metrizable if and only if there exists a metric $d$ on $S$ such that:

$\tau$ is the topology induced by $d$ on $S$.

Also see

• Definition:Metrizable Topology/Definition 2

• 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 British English spelling for metrizable is metrisable, but it is rarely found.

Sources

• 1955: John L. Kelley: General Topology: Chapter $4$: Embedding and Metrization
• 1970: Stephen Willard: General Topology: Chapter $2$: Topological Spaces: $\S3$: Fundamental Concepts: Example $3.1(\text a)$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metrizability
• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Topological Spaces: Topologies