Metrizable Space is Hausdorff

Theorem
Let $T$ be a metrizable topological space.

Then $T$ is a $T_2$ (Hausdorff) space.

Proof
By definition, $T$ is homeomorphic to a topological space $\left({S, \tau_d}\right)$ such that $\tau_d$ is the topology induced by a metric $d$ on $S$.

From Metric Space is Hausdorff, $\left({S, d}\right)$ is a $T_2$ (Hausdorff) space.

As the open sets of $\left({S, d}\right)$ are the same as the open sets of $\left({S, \tau_d}\right)$, it follows that $\left({S, \tau_d}\right)$ is a $T_2$ (Hausdorff) space.

From $T_2$ Space is Preserved under Homeomorphism it follows that $T$ is also a $T_2$ (Hausdorff) space.