Let $G$ be a connected Lie Group and $H$ a disconnected closed Lie Subgroup, defining $H_0$ as the connected component of the neutral element $1$ of $H$ we can define projection map
\begin{align} \pi:G/H_0&\to G/H\\ g H_0&\to g H. \end{align}
It is written in the book "Grupos de Lie - San Martin" that above map $\pi$ is a covering map, however, no demonstration is provided.
Does anyone know how to prove such a claim?
It is easy to see that $\pi$ is an open function and $\text{dim}(G/H_0) = \text{dim}(G/H)$. However, I don't know how to argue that this implies $\pi$ be a covering map.