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 if and only if $\left\{ {e}\right\}$ is closed in $G$.

Proof

Necessary Condition

Suppose $G$ is Hausdorff.

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

$\Box$

Sufficient Condition

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

Let $f: G \times G \to G$ be defined as:

$f \left({g h}\right) = g h^{-1}$

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

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

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

$\blacksquare$

Also see

• Topological Group is T1 iff T2
• Group is Hausdorff iff Discrete Subgroups are Closed