Fuzzy proximities and totally bounded fuzzy uniformities

Abstract

After the classical papers of Zadeh [ 181 and Chang [3 ], many concepts of general topology have been extended to fuzzy set theory: in this paper we are interested to develop the study of fuzzy uniformities, fuzzy proximities, and of the connexions between such structures. Fuzzy uniformities have been introduced by Lowen in [9] and, with slight modifications, in [lo] and by Hutton in [5]; the two approaches are quite different; however, the one proposed by Hutton seems to suit much better to fuzzy set theory, also because it allows to generalize a lot of results in the most proper way. Therefore in this paper we shall deal with Hutton fuzzy uniformities. The concept of fuzzy proximity used up to now (see, e.g., [ 7,8, 15 1) is quite unsatisfactory: indeed its “fuzzyness ” is rather poor since these fuzzy proximities are in a canonical l-1 correspondence with the usual proximities (see Remark 2.6): moreover the open sets of the induced topologies are crisp and, though every Lowen fuzzy uniformity induces a fuzzy proximity, this correspondence cannot work well since the two structures do not give the same fuzzy topology. For these reasons we propose a new definition of fuzzy proximity; it differs from the old one only in Axiom P5: in short we could say that we privilege complementation with respect to intersection (an agreement between the two operations being problematic in fuzzy set theory). We point out that, as a consequence, a fuzzy set may be “far” from itself: however, this fact ought not to amaze anyone since it may happen just for “pale” sets which, in a certain sense, resemble the empty set. Nevertheless, this modification enables us to associate a topology in a completely different way; moreover, every fuzzy uniformity induces a fuzzy proximity and, vice versa, we succeed to construct a fuzzy uniformity starting with a given fuzzy proximity: and the induced topologies do not change at any step. This implies that the topologies which admit a fuzzy proximity are exactly the completely regular ones. Moreover, we prove that there exists a l-l correspondence between fuzzy proximity spaces and a