SMOOTH PREUNIFORM AND PREPROXIMITY SPACES

Fuzzy sets and fuzzy logic are very suited to interpret every day life sentences. Fuzzy interpretations are defined with the use of membersh ip functions whose role is simply to give a kind of graphical, visual support to help us to capture the meaning. We think that generally mem bership values are not very important, but only some characteristics o f membership functions have to be kept. We think that monotonic relati ons between degrees with which some properties are satisfied is an imp ortant characteristic. We translate in this sense axioms corresponding to preuniform structures and preproximity structures. Contrary to fuz zy extensions, like fuzzy topology for instance, which generally keep the original axioms but adapt few of them to deal with fuzzy sets, our approach induce much more transformations. Finally we study how class ic constructions relating these concepts can be translated in this fra mework.