Topological Group is T1 iff T2

Theorem

Let $G$ be a topological group.

Then $G$ is a $T_1$ space if and only if $G$ is Hausdorff.

Proof

Necessary Condition

Follows directly from $T_2$ Space is $T_1$ Space.

$\Box$

Sufficient Condition

Let $G$ be a $T_1$ space.

Then $\left\{ {e}\right\}$ is closed.

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

$\blacksquare$