Homeomorphism Between Fuzzy Number Space and the Space of Bounded Functions with Same Monotonicity on $$[-1,1]$$ [ - 1 , 1

Abstract

In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space $$\varepsilon ^1$$e1 and the space $$B[-1, 1]$$B[-1,1], which defined as a set of standard monotonic bounded functions with monotonicity on interval $$[-1, 1]$$[-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 $$B[-1, 1]$$B[-1,1], Hausdorff metric and $$L_p$$Lp metric, which both are classical functional metrics, is adopted and their topological properties is discussed. In addition, by the means of introducing fuzzy functional to space $$B[-1, 1]$$B[-1,1], we present two new fuzzy number's metrics. Finally, according to the proof of homeomorphism between fuzzy number space $$\varepsilon ^1$$e1 and the space $$B[-1, 1]$$B[-1,1], it's argued that not only it gives a new way to study the fuzzy analysis theory, but also make the study of fuzzy number space easier.