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 .