# Definition:Metrizable Topology

## 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

• Results about metrizable topologies can be found here.

## Linguistic Note

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