A characterization of the circle group via uniqueness of roots

作者：Dikranjan Dikran^*; Di Santo Raffaele

来源：Topology and Its Applications, 2011, 158(2): 159-166.

DOI：10.1016/j.topol.2010.08.014

• Locally compact group
• Circle group
• Topologically torsion element
• Sequential limit law
• Characterizing sequence

It was shown in Biro 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(n)x -> 0 for some x is an element of T iff x is an element of 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(n)x -> 0 in G for all sequences (u(n)) of integers such that u(n)h -> 0 in G for all h is an element of 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 congruent to T.

• 出版日期2011-2-1