The space of locally elliptic subgroups of a locally compact group

Let G be a locally compact group. We denote by Sub(G) the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper we establish that the subset SubLE(G) of closed locally elliptic subgroups of G is Chabauty-closed if and only if the set compn(G) consisting of the n-tuples of elements of G contained in a common compact subgroup, is closed in the Cartesian product Gn, for every nonnegative integer n. Moreover, we prove that the subspace SubLE(G) is Chabauty-closed in the case when the group G contains an open normal locally elliptic subgroup.