FUZZIFYING CLOSURE SYSTEMS AND CLOSURE OPERATORS

Abstract

In this paper, we propose the concepts of fuzzifying closure systems <br />and Birkhoff fuzzifying closure operators. In the framework of fuzzifying <br />mathematics, we find that there still exists a one to one correspondence between <br />fuzzifying closure systems and Birkhoff fuzzifying closure operators as <br />in the case of classical mathematics. In the aspect of category theory, we prove <br />that the category of fuzzifying closure system spaces is isomorphic to the category <br />of Birkhoff fuzzifying closure spaces. In addition, we obtain an important <br />result that the category of fuzzifying closure spaces and that of fuzzifying closure <br />system spaces can be both embedded in the category of Birkhoff <br /><em><span style="font-size: xx-small; font-family: CMMI8;"><em><span style="font-size: xx-small; font-family: CMMI8;">𝐼</span></em></span></em> <br /><span style="font-size: xx-small; font-family: CMR8;">-closure</span> <br />spaces. Finally, using fuzzifying closure systems of the paper, we introduce a <br /> <br />set of separation axioms in fuzzifying closure system spaces, which offer a try <br /> <br />how to research the properties of spaces by fuzzifying closure systems. <br /> <br />