Duality properties of bounded torsion topological abelian groups

Let G be a precompact, bounded torsion abelian group and Gp∧ its dual group endowed with the topology of pointwise convergence. We prove that if G is Baire (resp., pseudocompact), then all compact (resp., countably compact) subsets of Gp∧ are finite. We also prove that G is pseudocompact if and only if all countable subgroups of Gp∧ are closed. We present other characterizations of pseudocompactness and the Baire property of Gp∧ in terms of properties that express in different ways the abundance of continuous characters of G. Besides, we give an example of a precompact boolean group G with the Baire property such that the dual group Gp∧ contains an infinite countably compact subspace without isolated points.