Metrizable Space is Hausdorff

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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 $\struct {S, \tau_d}$ such that $\tau_d$ is the topology induced by a metric $d$ on $S$.

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

As the open sets of $\struct {S, d}$ are the same as the open sets of $\struct {S, \tau_d}$, it follows that $\struct {S, \tau_d}$ 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.

$\blacksquare$


Sources