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 $\operatorname{im} \left({f}\right)$ be Hausdorff.

Then its kernel $\ker \left({f}\right)$ is closed in $G$.

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

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

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

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