Definition:Completely Metrizable Topology

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

The space $\struct {S, \tau}$ is said to be completely metrizable 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.