Fuzzy preuniform structures and the structures they induce. 1. Main results

Abstract

With the increasing complexity of the systems that men have created or have tried to understand, it has appeared that the classic quantitative tools are not well adapted. We must take into account the imprecision inherent to every modelling of complex systems. On this problem, Zadeh [ 131 stated the following principle: “As the complexity of a system increases our ability to make precise and yet significant statements about its behaviour diminishes, until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics.” For this reason qualitative tools have been developed: topological and connected theories, qualitative dynamics, etc., and new theories have been built up: fuzzy sets and possibility theories. In fact, the modelling problem is a continuity-like one, in the sense that it must carry and preserve certain important features of the original real world. It is certainly this problem that Thorn has in mind when he says that qualitative dynamics “consiste a admettre a priori l’existence d’un modele differentiel sous-jacent au processus etudie et, faute de connaitre explicitement ce modele, a deduire de la seule supposition de son existence des conclusions relatives a la nature des singularitis du processus” [ 121. In this paper we generalize the concept of uniform space by defining preuniform spaces and a typology on them. At the same time we bring this theory together with that of fuzzy sets in order to get a very powerful tool. We have used the Zadeh definition of fuzzy sets, but the major part of our results can be translated into the case of L-fuzzy sets when