Topological Group is Hausdorff iff Identity is Closed

Theorem
Let $G$ be a topological group.

Let $e$ be its identity element.

Then $G$ is Hausdorff $\{e\}$ is closed.

Necessary Condition
Suppose $G$ is Hausdorff.

By T2 Space is T1 Space, $\{e\}$ is closed.

Sufficient Condition
Suppose $\{e\}$ is closed.

Let $f:G\times G\to G$ be defined as $f(gh)=gh^{-1}$.

Because $f$ is continuous, $f^{-1}(e)$ is closed.

We have that $f^{-1}(e)$ is the diagonal set on $G$.

By Hausdorff Space iff Diagonal Set on Product is Closed, $G$ is Hausdorff.

Also see

 * Group is T1 iff T2
 * Group is Hausdorff iff Discrete Subgroups are Closed