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This paper constructs a counterexample showing that not every locally$L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms.Random convex analysis is the analytic foundation for $L^0$--convex conditionalrisk measures, this counterexample, however, shows that a locally $L^0$--convexmodule is not a proper framework for random convex analysis. Further, thispaper also gives a necessary and sufficient condition for a locally$L^0$--convex topology to be induced by a family of $L^0$--seminorms. Finally,we give some comments showing that based on random locally convex modules, wecan establish a perfect random convex analysis to meet the needs of the studyof $L^0$--convex conditional risk measures.

A counterexample shows that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms