Although this is quite a classical result, I really like it.

**Lemma.*** Let be a topological group. Then is if and only if is Hausdorff.*

*Proof.* One implication is clear. Conversely, suppose is . Then the identity element is closed. The map

is continuous. Hence, the inverse image of the identity is closed. But this is the diagonal, hence is Hausdorff.

**Exercise.** Prove that Hausdorff is in fact equivalent to .

Nice! I knew this result for topological vector spaces. Didn’t realise it holds for any topological group.