LAUSR

Abstract Let G be a locally compact topological group, G 0 the connected component of its identity element, and comp ( G ) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB ( G ) of all closed subgroups of G carries a compact Hausdorff topology called the Chabauty topology. Let F 1 ( G ) , respectively, R 1 ( G ) , denote the subspace of all discrete subgroups isomorphic to Z , respectively, all subgroups isomorphic to R . It is shown that a necessary and sufficient condition for G ∈ F 1 ( G ) ‾ to hold is that G is Abelian, and either that G ≅ R × comp ( G ) and G / G 0 is inductively monothetic, or else that G is discrete and isomorphic to a subgroup of Q . It is further shown that a necessary and sufficient condition for G ∈ R 1 ( G ) ‾ to hold is that G ≅ R × C for a compact connected Abelian group C .