The torsion Bohr topology for Abelian groups

For an abstract Abelian group G, denote by G ♮ the group G endowed with the torsion Bohr topology induced by all (abstract) homomorphisms of G to the torsion subgroup, tor ( T ) , of the circle group T . Among the main results of this article are: A homomorphism χ : G ♮ → T is continuous if and only if χ ( G ) ⊆ tor ( T ) . Every compact subset of G ♮ is countable. A subset B ⊂ G ♮ is bounded if and only if c l G ♮ B is compact and countable. Furthermore, χ ( G ♮ ) = ω if and only if G is finitely generated. Also we prove that Z ♮ is metrizable non-discrete with w ( Z ♮ ) = ω and Q ♮ is non-metrizable with w ( Q ♮ ) = c , where Z and Q are the groups of integer and rational numbers, respectively.