Pontryagin duality in the class of precompact Abelian groups and the Baire property

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements. They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G∧ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group G of weight ≥2ω, we find proper dense subgroups H1 and H2 of G such that H1 is reflexive and pseudocompact, while H2 is non-reflexive and almost metrizable.