# Definition:Metrizable Topology

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## Definition

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

### Definition 1

$T$ 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 $S$.

### Definition 2

$T$ 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

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

- Results about
**metrizable topologies**can be found**here**.

## Linguistic Note

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

## Sources

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- 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