The Mackey problem for free abelian topological groups

The classical Mackey–Arens theorem states that every locally convex space has a Mackey space topology. However, in the wider class of locally quasi-convex (lqc) groups an analogous result does not hold. Indeed, Außenhofer and the author showed independently that the free abelian topological group A(s) over a convergent sequence s does not admit a Mackey group topology. We essentially extend this example by showing that the free abelian topological group A(X) over a non-discrete zero-dimensional (for example, countable) metrizable space X does not have a Mackey group topology.