Fuzzy Closure Hopf Space

In this study, we introduce the concepts of fuzzy closure Hopf space and fuzzy closure Hopf group within the framework of fuzzy closure spaces, using homotopy theory. We investigate the relationships between the fuzzy closure Hopf group and its homotopy equivalence. Furthermore, we demonstrate the existence of a contravariant functor from the category of fuzzy closure Hopf spaces and the continuous functions, to the category of groups and homomorphisms. This is demonstrated by illustrating that the set of homotopy function classes among fuzzy closure Hopf groups constitutes a group.