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 ) .