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 Hausdorff Condition is Preserved under Homeomorphism it follows that $T$ is also a $T_2$ (Hausdorff) space.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Exercise $1$