A characterization of the circle group via uniqueness of roots

It was shown in Bíró et al. (2001) [7] that every cyclic subgroup C of the circle group T admits a characterizing sequence (un) of integers in the sense that unx→0 for some x∈T iff x∈C. More generally, for a subgroup H of a topological (abelian) group G one can define:

(a)

g(H) to be the set of all elements x of G such that unx→0 in G for all sequences (un) of integers such that unh→0 in G for all h∈H;

(b)

H to be g-closed if H=g(H).

We show then that an infinite compact abelian group G has all its cyclic subgroups g-closed iff G≅T.