On Metric Properties Of Fuzzy Number Spaces

In this thesis,we investigate the metric properties of fuzzy number spaces.The content of this thesis are as followings:1.First,it is proved that the endogragh metric is approximative with respect to orders on noncompact fuzzy number space E^.It is also shown that the endograph metric approach on orders on E is constructive,this shows that noncompact fuzzy-number-valued integrals such as M-integral and G-integral are computable.Finally some analytic properties with respect to the endograph metric on E are given.2.The endograph and sendograph metrics on the spaces of fuzzy numbers are known to be separable,but neither of them is complete.This paper deals with the completions of the endograph and sendograph metrics.It is proved that E and {([a,b]×{0})∪send(u):u₀(?)[a,b],u(?)E¹} are the completions of E¹ with respect to the endograph and sendograph metrics respectively.The completion of E¹ with respect to the endograph metric enables us for the first time to consider a separable and complete metric on E^, the space of noncompact fuzzy numbers;While in considering completion with respect to the sendograph metric,a uniformly equivalent description of the sendograph metric is given which reveals that there are some internal relations between the sendograph and endograph metrics.3.It is proved that fuzzy numbers can be approximated via sendograph metric by fuzzy numbers with continuousλ-cut functions to any accuracy.Our methods of approximation is finite in nature.Relations between fuzzy numbers with Zadeh's extension principle are discussed.