On the Characterization of Locally L^(0)-Convex Topologies Induced by a Family of L^(0)-Seminorms

The purpose of this paper is to provide a characterization of locally L0-convex modules induced by a family of L0-seminorms using the gauge function for L0-modules and discussing the notion of countable concatenation. Taking advantage of these ideas we will give a counterexample of a locally L0-convex topology not induced by any family of L0-seminorms.