Locally compact groups approximable by subgroups isomorphic to Z or R

Let G be a locally compact topological group, G0 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 F1(G), respectively, R1(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∈F1(G)‾ to hold is that G is Abelian, and either that G≅R×comp(G) and G/G0 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∈R1(G)‾ to hold is that G≅R×C for a compact connected Abelian group C.