If $G$ is a Hausdorff, locally compact topological group, and $H$ is a closed subgroup of $G$ then is it true that all connected components of $G/H$ are the closure of the image of connected components of $G$ under quotient map $p:G\to G/H$?
I want to show any $g\in G$, the image of connected component of $g$ under $p$ is dense in the connected component of $p(g)$ but don't know how to do it?