Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1]

In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space e1 and the space B[−1, 1], which defined as a set of standard monotonic bounded functions with monotonicity on interval [−1, 1]. Furthermore, a new approach based upon the monotonic bounded functions has been proposed to create fuzzy numbers and represent them by suing fuzzy structured element. In order to make two different metrics based space in B[−1, 1], Hausdorff metric and Lp metric, which both are classical functional metrics, are adopted and their topological properties are discussed. In addition, by the means of introducing fuzzy functional to space B[−1, 1], we present two new fuzzy number’s metrics. Finally, according to the proof of homeomorphism between fuzzy number space e1 and the space B[−1, 1], it’s argued that not only does it give a new way to study the fuzzy analysis theory, but also makes the study of fuzzy number space easier.