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:
 * $\forall s,t \in S: \map {d_\phi} {s,t} = \map d {\map \phi s, \map \phi t}$

$d_\phi$ satisfies $(\text M 1)$
We have:

Hence $d_\phi$ satisfies metric axiom $(\text M 1)$.

$d_\phi$ satisfies $(\text M 2)$
We have:

Hence $d_\phi$ satisfies metric axiom $(\text M 2)$.

$d_\phi$ satisfies $(\text M 3)$
We have:

Hence $d_\phi$ satisfies metric axiom $(\text M 3)$.

$d_\phi$ satisfies $(\text M 4)$
By definition of homeomorphism:
 * $\forall s, t \in S : s \ne t \iff \map \phi s \ne \map \phi t$

We have:

Hence $d_\phi$ satisfies metric axiom $(\text M 4)$.

It follows that $d_\phi$ is a metric on $S$ by definition.

Lemma 2

 * $\forall s \in S, \epsilon \in \R_{\ge 0} : \map {B_\epsilon} s = {\phi^{-1}} \sqbrk {\map {B_\epsilon} {\map \phi s}}$

where
 * $(1) \quad \map {B_\epsilon} s$ is the open ball in $\struct{S, d_\phi}$ with center $s$ and radius $\epsilon$
 * $(2) \quad \map {B_\epsilon} {\map \phi s}$ is the open ball in $\struct{A, d}$ with center $\map \phi s$ and radius $\epsilon$

Lemma 1

 * $\forall U \subseteq S : U$ is open in $\struct{S, d_\phi}$ $\phi \sqbrk U$ is open in $\struct{A, d}$

Let $\tau_\phi$ be the topology induced by the metric $d_\phi$.

We have:
 * $U$ is open in $\struct{S, \tau}


 * $\phi \sqbrk U$ is open in $\struct{A, \tau_d} //Homeomorphism


 * $\phi \sqbrk U$ is open in $\struct{A, d} //Topology induced by metric


 * $U$ is open in $\struct{S, d_\phi} //Lemma 2

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