FUZZIFYING CLOSURE SYSTEMS AND CLOSURE OPERATORS

In this paper, we propose the concepts of fuzzifying closure systems and Birkhoff fuzzifying closure operators. In the framework of fuzzifying mathematics, we find that there still exists a one to one correspondence between fuzzifying closure systems and Birkhoff fuzzifying closure operators as in the case of classical mathematics. In the aspect of category theory, we prove that the category of fuzzifying closure system spaces is isomorphic to the category of Birkhoff fuzzifying closure spaces. In addition, we obtain an important result that the category of fuzzifying closure spaces and that of fuzzifying closure system spaces can be both embedded in the category of Birkhoff I-closure spaces. Finally, using fuzzifying closure systems of the paper, we introduce a set of separation axioms in fuzzifying closure system spaces, which offer a try how to research the properties of spaces by fuzzifying closure systems.