A proper subgroup H of an
LCA group G is totally dense (briefly, t.d.) in G if H ∩ K is dense in K for every
closed subgroup K of G. Let B(G) and t(G) denote, respectively, the subgroup of
compact elements of G and the maximal torsion subgroup of G. We say that G
is an admissible group if G = B(G)≠t(G). An element g ∈ G is called a
metric element of G if ⟨g⟩, the closure of the cyclic subgroup generated by G,
is metrizable. The following are the main results of this paper. (1) Let G
be an admissible LCA group such that G0, the component of the identity
in G, is nonmetrizable. Then m(G), the set of all metric elements of G,
is a t.d. subgroup of G. (2) An LCA group G contains a t.d. subgroup if
and only if G is an admissible group. (3) A characterization of those LCA
groups is given for which the maximal torsion subgroups are t.d. in the
groups.
