Polytopic and TS models are nowhere dense in the approximation model space

We show in this paper that the set of functions, consisting of polytopic or TS models constructed from finite number of components, is nowhere dense in the approximation model space, if that is defined as a subset of continuous functions. This topological notion means that the given set of functions lies “almost discretely” in the space of approximated functions. As a consequence, by means of the mentioned models we cannot approximate in general continuous functions arbitrarily well, if the number of components are restricted. Thus, only functions satisfying certain conditions can be approximated by such models, or alternatively, we need unbounded number of components. The possible solutions are outlined in the paper. Keywords— TS model, polytopic model, function approximation.