Hello,
Let $G$ denote a locally compact group and $S(G)$ the chabauty space of $G$, that is the set of closed subgroups of $G$ equiped with the chabauty topology, it is a compact space.
My question is : if $G$ is a locally profinite group, for example $G=GL(n,F)$ where $F$ is a non-archimidian field, and $\mathcal{OK}(G)$ the set of open compact subgroup of $G$, it is true that $\mathcal{OK}(G)$ is closed in $S(G)$ ?