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A set of time dependent fuzzy membership functions and its compactness

By regarding the set of fuzzy membership functions as a subset of the set of functions which depend on time, we show compactness in weak topology. We prove also that the function which composes the control of the function given by the center of gravity calculation is continuous for the topology of pointwise convergence. As an application, we discuss the existence of an optimal solution in fuzzy control for an open-loop system.