  The purpose of this paper is to provide a characterization of the topological
$L^0$-modules whose topology is induced by a family of $L^0$-seminorms using
the gauge function for $L^0$-modules. Taking advantage of these ideas we will
give a counterexample of a locally $L^0$-convex module whose topology is not
induced by a family of $L^0$-seminorms.
