I'm trying to show that if $G$ is a Lie group, then it is locally connected, i.e., for any point $p$ in an open subset $U$ of $G$, there is a connected neighborhood $V$ of $p$ such that $V\subset U$. This is a step in my proof that the identity component of a Lie group is open, since I know that the connected components of a locally connected topological space are open.
My thought was to say let $U$ be an open subset of $G$ and let $g\in U$. Then letting $G_0$ be the connected component of the identity, since left-multiplication by $g$ is a smooth map from $G$ to itself, I know that $gG_0$ is a connected component of $G$ containing $g$. However, is $gG_0 \subset U$?
