Sugeno controllers with a bounded number of rules are nowhere dense

In literature various results can be found claiming theat fuzzy controllers are universal approximators. In terms of topology this means that fuzzy controllers as subsets of adequate function spaces are dense. In this paper the topological structure of fuzzy controllers composed of a bounded number of rules is investigated.It turns out that these sets are nowhere dense (a topological notion indicating that the sets are "almost discrete"). This means, that it is just the number of rules and, e.g., not the great variety of parameters of fuzzy controllers, why fuzzy controllers are universal approximators.