  In his PhD Thesis Konstantinos Beros proved a number of results about
compactly generated subgroups of Polish groups. Such a group is K-sigma - 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
K-sigma (since it is countable) but not compactly generated. Beros showed that
for any Polish group G, every K-sigma subgroup of G is compactly generated iff
every countable subgroup of G is compactly generated. Beros showed that any
K-sigma subgroup of Z^omega (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 G^omega
is compactly generated. In unpublished work Beros asked whether finitely
generated may be replaced by compactly generated in his theorem. 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 countable subgroup of
R^omega is compactly generated. We prove 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 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 R^n which meets every (n-1)-dimensional
subspace in a discrete set. Similarly there is a dense subgroup of R^omega
which meets every finite dimensional subspace of R^omega in a discrete set.
