(Some notations:)
I'm a bit confused by the definition support in the compact induced representation:
It says $f_w$ is supported in $H$, so I assume this means the set supp($f$)= closure of { $x$; $x\in G$, $f(x)\neq0$ } is in $H$. But when $w\neq0$, $f_w(h)\neq0$ for all $h\in H$, so supp($f$) is the closure of $H$, its closure should be bigger than itself unless $H$ is closed, but $H$ is open by assumption, This makes me extremely confused.
In a topological group, open subgroups are closed. Since $G$ is the union of its $H$-cosets, and cosets of $H$ are open. So $G\backslash H$ is the union of a set of opens, hence open, hence $H$ is closed.