Let $G$ be a topological group with identity $e$ such that $\left\{e\right\}$ is closed. Then $G$ is $T_0$ and regular.
$G$ is trivially $T_0$.
Let $C\subset G$ be closed and $C\not\ni g\in G$. For every $h\in C$, I was able to find an open neighborhood $U_h\not\ni g$ of $h$. I am stuck trying to find an open neighborhood $U$ of $g$ disjoint from $\bigcup_{h\in C}U_h$.