Fuzzy Numbers And Fuzzy Metric Space

In this dissertation, we investigate the theory of fuzzy number space and fuzzy metric space, the main works of this dissertation are as following:1 . We introduce the notion of dais point of fuzzy number net and use it to characterize the level convergence given by Kaleva and Seikkla [54]. Then we use the local subbase to give the structure of topology r(l) induced by level convergence and prove that the space (En, r(l)) is a T% space satisfying the first countability axiom. The notion of eventually equi-left (right)-continuity is introuduced, by making use of it, we obtain an sufficient and necessary condition under which the limit of a sequence of fuzzy numbers exists in the sense of level convergence. On this basis, we give the characterization of the compact set in (En, r(l)), i.e. a closed subset U of (E", r(l)) is compact iff U is uniformly support-bounded and each net in U has a subnet which is eventually equi-left-coutinuous on (0,1] and eventually equi-left-coutinuous at A = 0.2. We give a brief proof of the existence theorem of Supremum and Infimum of a bounded set of fuzzy numbers given by Wu cong-xin and Wu chong in [30]. It is used to establish the monotone convergence theorem and the nest theorem of closed intervals on (En, t(l))3. Using the equivalent condition under which the limit of a sequence of fuzzy numbers exists in the sense of level convergence, we analyze the structure of the level continuous fuzzy number valued function. On the basis of it, we prove the level continuous fuzzy number valued function on a closed interval exists supremum and infimnm and give the precise representation. Since r(l) is weaker than the topology induced by the metric doo on El, this conclusion essentially extends the results in [30].4. We give an equivalent characterization of the metric D^ and investigate the properties of the sendgraph of fuzzy numbers. On this basis, we show the assertion A[a, 6] B[a, b] of Buckley etc is incorrect by giving a counterexample. Where ,A[a, b] and B[a, b] denote the space of all continuous functions from El [a, 6] to El with respect to the metrics D and dt?respectively, El[a, b] denotes the set of all one-dimensional fuzzy numbers with support in [a,b].5. Under more ordinary conditions of L and R, we give an equivalent characterization of triangle inequality (iii) in definition of fuzzy metric space. It extends the result obtained by Kaleva and Seikkla [54] under the condition that L = min, R = max. Further, surmounting the confine of L = min and R = max, we establish a general completion theorem of fuzzy metric spaces.