  In this article it is shown that to every non-discrete Hausdorff linear
topology on $\Z$ other metrizable locally quasi-convex group topologies can be
associated which are strictly finer than the linear topology and such that the
character groups coincide. Applying this result to the $p$-adic topology on
$\Z$, we give a negative answer to the question of Dikranjan, whether this
topology is Mackey.
