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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 usingthe gauge function for $L^0$-modules. Taking advantage of these ideas we willgive a counterexample of a locally $L^0$-convex module whose topology is notinduced by a family of $L^0$-seminorms.

On characterization of locally $L^0$-convex topologies induced by a family of $L^0$-seminorms