Image of Group Homomorphism is Hausdorff Implies Kernel is Closed

Theorem
Let $G$ and $H$ be topological groups.

Let $f:G\to H$ be a morphism.

Let its image $\operatorname{im}(f)$ be Hausdorff.

Then its kernel $\ker(f)$ is closed in $G$.

Proof
By Image of Group Homomorphism is Subgroup, $\operatorname{im}(f)$ is a group.

Let $e$ be the identity of $H$.

By Group is Hausdorff iff Identity is Closed, $\{e\}$ is closed in $\operatorname{im}(f)$.

Because $f$ is continuous, $\ker(f) = f^{-1}(e)$ is closed in $G$.