Locally Compact Groups with Compact Open Subgroups Having Open Chabauty Spaces

Let G be a locally compact group. We denote by SUB (G) the space of closed subgroups of G equipped with the Chabauty topology; this is a compact space. The topological space SUB (G) is called the Chabauty space of G. For a closed subgroup H of G the subspace L SUB (G) | L H of SUB (G) is homeomorphic to the Chabauty space SUB (H) of H and so SUB (H) is a compact subspace of SUB (G). The paper discusses the scope of validity of an assertion having appeared recently in the book of Herfort-Hofmann-Russo about the openness of the subspace SUB (H) in SUB (G). We study the class X of locally compact groups G such that the subspace SUB (H) is open in SUB (G) for any compact open subgroup H of G. We show that a locally compact abelian group A is in X if and only if A contains a compact open subgroup U such that A/U is a finite direct sum of subgroups each of which is either cyclic or is a Prfer group isomorphic to Z (p).