A CHARACTERIZATION OF 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 (u_n) of integers in the sense that u_nx → 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 u_nx → 0 in G for all sequences (u_n) of integers such that u_nh→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.