Separation properties for topological groups

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

Lemma. Let G be a topological group. Then G is T_1 if and only if G is Hausdorff.

Proof. One implication is clear. Conversely, suppose G is T_1. Then the identity element is closed. The map

    \begin{align*} G\times G &\rA G\\ (g,h) &\rM gh^{-1} \end{align*}

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

Exercise. Prove that Hausdorff is in fact equivalent to T_0.