# Definition:Completely Metrizable Topology

## Definition

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

The space $\struct {S, \tau}$ is said to be **completely metrizable** if and only if there exists a metric $d$ such that:

- $\struct {S, d}$ is a complete metric space

and:

- $\tau$ is the topological space induced by the metric $d$.

## Also see

All completely metrizable topologies are in particular metrizable topologies.