Q is not a Mackey group

Abstract

The aim of this paper is to prove that the usual topology in Q inherited from the real line is not a Mackey topology in the sense defined in [5]. To that end, we find a locally quasi-convex topology on Q/Z, the torsion group of T, which is strictly finer than the one induced by the euclidean topology of T. Nevertheless, both topologies on Q/Z admit the same character group. Since the property of being a Mackey group is preserved by LQC quotients, we obtain that the usual topology in Q is not the finest compatible topology. In other words, there is a strictly finer locally quasi-convex topology on Q giving rise to the same dual group as Q with the usual topology. A wide class of countable subgroups of the torus T, which are not Mackey are also obtained ( Remark 3.7). Obviously, they are precompact, metrizable and locally quasi-convex groups.