The torsion Bohr topology for Abelian groups

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