Topological Group is T1 iff T2

Theorem
Let $G$ be a topological group.

Then $G$ is a $T_1$ space iff $G$ is Hausdorff.

Proof
If $G$ is Hausdorff, it follows by T2 Space is T1 Space that $G$ is $T_1$.

Suppose $G$ is $T_1$. Then $\{e\}$ is closed.

By Group is Hausdorff iff Identity is Closed, $G$ is Hausdorff.