Why is the quotient of a group over its connected component of the identity, $H/H^0$ discrete?

Question

Let $G$ be a Lie group and $H$ is a closed subgroup. Why $H/H^0$ is a discrete Lie group, where $H^0$ is the connected component containing the identity of $H$?

Answer 1

Consider $p:G\rightarrow G/G^0$ be the quotient map. $p$ is an open map since $G/G^0$ is endowed with the quotient topology, it implies that for every $g\in G$, $p(gG^0)=p(g)$ is open. It implies that every element of $G/G^0$ is an open subset. If $H$ is a closed subgroup of $G$, it is a Lie group, so the argument above can be applied.