On the existence of topologies compatible with a group duality with predetermined properties

The paper deals with group dualities. A duality is simply a pair (G,H) where G an abstract abelian and H subgroup of characters defined on G. topology τ compatible the (also called dual pair) if equipped has H. topological (G,τ) gives rise to natural (G,G∧), G∧ stands for continuous We prove that existence g-barrelled (G,G∧) equivalent semireflexivity in Pontryagin's sense endowed pointwise convergence σ(G∧,G). also deal k-group topologies. topologies determined by sort completeness property its Bohr σ(G,G∧) (Theorem 3.3).