The Research Of Fuzzifying Closure Systems

Closure systems and closure operators play a significant role in classicalmathematics, involving kinds of di?erent domains. This paper studies fuzzifyingclosure systems and Birkho? fuzzifying closure operators, and investigates theproperties of fuzzifying closure system spaces.This paper is organized as follows:The first section is preface. We introduce the development of closure systemsand closure operators, and propose the main problems of this paper.The second section is about investigation of fuzzifying closure systems andBirkho? fuzziying closure operators. In this part, we propose the concepts ofthem in fuzzifying cases. In the framework of fuzzifying mathematics, we findthat there exists a one to one correspondence between fuzzifying closure systemsand Birkho? fuzzifying closure operators as in the case of classical mathematics.In the aspect of category theory, we prove that the category of fuzzifying closuresystem spaces is isomorphic to the category of Birkho? fuzzifying closure spaces.In addition, we also prove that the category of fuzzifying closure spaces and thatof fuzzifying closure system spaces can be both embedded in the category ofBirkho? I-closure spaces. At the end of this part, we obtain the further resultsabout the properties of fuzzifying closure system spaces and the sub-fuzzifyingclosure system spaces.The third section investigates the separation axioms of fuzzifying closuresystem spaces. We establish a kind of fuzzifying neighborhood structures of afuzzifying closure system space, called a fuzzifying remote neighborhood systemwhich is used for studying the local problems of fuzzifying closure system spaces,and then several separation axioms claimed are established. Moreover, the rela-tions among these separation axioms are investigated.