Equivalence of Definitions of Metrizable Topology

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

Definition 1 implies Definition 2
Let $d$ be a metric on $S$ such that $\tau$ is the topology induced by $d$.

From Identity Mapping is Homeomorphism:
 * $T$ is homeomorphic to a topological space with a topology induced by a metric.

Definition 2 implies Definition 1
Let $M = \struct{A, d}$ be a metric space such that $T$ is homeomorphic to $\struct{A,\tau_d}$ where $\tau_d$ is the topology induced by $d$.

Let $\phi : \struct{S, \tau} \to \struct{A, \tau_d}$ be a homeomorphism.

Let $d_\phi : S \times S \to \R_{\ge 0}$ be the mapping defined by:
 * $\forall s,t \in S: \map {d_\phi} {s,t} = \map d {\map \phi s, \map \phi t}$

Lemma 4
It remains to show that $\tau_\phi$ is the topology induced by the metric $d_\phi$.

We have:

Hence $\tau$ is a topology induced by a metric by definition.