I read a theorem that if $G$ is a locally connected group, then the component of the identity $G_0$ is generated by any connected neighborhood of $e$.
It goes like: Let $V$ be a connected nbhd of $e$, $H$ the generated subgroup of $V$. For any $g\in H$, $gV$ is a connected nbhd of $g$ in $H$, so $H$ is open. Thus $H$ is closed as an open subgroup of a topological group. Since $H$ is closed and open, it is the union of the connected components of its points.
So since $e\in H$, $G_0\subseteq H$. Why should equality follow though? This is Theorem 10.21 in Peter Szekeres' A Coure in Modern Mathematical Physics.