Definition:Metrizable Topology/Definition 2

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

$T$ is said to be metrizable 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

 * Definition:Metrizable Topology/Definition 1


 * Equivalence of Definitions of Metrizable Topology


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


 * Definition:Completely Metrizable Topology