In this note, we prove that in a Hausdorff topological abelian group, the closed subgroup generated by all compact elements is equal to teh closed subgroup generated by all compact elements which are topologically p-torsion for some prime p. In particular, this yields a new, short solution to a question raised by Armacost [A]. Using Pontrjagin duality, we obtain new descriptions of the identity componant of a locally compact abelian group.
Loth, Peter. "Compact Topologically Torsion Elements of Topological Abelian Groups." Rendiconti del Seminario Matematico della Università di Padova 113 (2005): 117-123.