Countable subgroups of Euclidean space

In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is Kσ, the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is Kσ (since it is countable) but not compactly generated (since compact subsets are finite). Beros showed that for any Polish group G, every Kσ subgroup of G is compactly generated iff every countable subgroup of G is compactly gener-ated. He showed that any countable subgroup of Zω (infinite product of the integers) is compactly generated and more generally, for any Polish group G, if every countable subgroup of G is finitely generated, then every countable subgroup of Gω is compactly generated. In unpublished work Beros asked the question of whether “finitely gen-erated ” may be replaced by “compactly generated ” in this last result. He conjectured that the reals R under addition might be an example such that every countable subgroup of R is compactly generated but not every count-able subgroup of Rω is compactly generated. We prove (Theorem 4) that this is not true. The general question remains open. In the course of our proof we came up with some interesting countable subgroups. We show (Theorem 6) that there is a dense subgroup of the plane which meets every line in a discrete set. Furthermore for each n there is a dense subgroup of Euclidean space Rn which meets every n − 1 dimensional subspace in a discrete set and a dense subgroup of Rω which meets every finite dimensional subspace of Rω in a discrete set. Theorem 1 Every countable subgroup G of R is compactly generated. Proof If G has a smallest positive element, then this generates G. Otherwise let xn ∈ G be positive and converge to zero. Let G = {gn: n < ω}. For each n choose kn ∈ Z so that |gn − knxn | ≤ xn. Let C = {0} ∪ {xn, gn − knxn: n < ω} 1 then C is a sequence converging to zero, so it is compact. Also gn = (gn − knxn) + knxn so it generates G. QED Theorem 2 For 0 < m < ω every countable subgroup G of Rm is compactly generated. Proof For any > 0 let V = spanR({u ∈ G: ||u| | < }) where here the span is taken respect to the field R. Note that for 0 < 1 <