  Let a group $\Gamma$ act on a paracompact, locally compact, Hausdorff space
$M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We
endow $2^M$ with the Chabauty topology, which is compact and admits a natural
$\Gamma$-action by homeomorphisms. We show that for every minimal
$\Gamma$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact
sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure.
  As an application, we deduce that every compact uniformly recurrent subgroup
of a locally compact group is contained in a compact normal subgroup. This
generalizes a result of U\v{s}akov on compact subgroups whose normalizer is
compact.
