On Quasi-discrete Fuzzy Closure Spaces

This paper studies quasi-discrete closure spaces and fuzzy closure spaces. We show that any topological closure c_T induced by a closure c is the smallest extension from a closure space to a topological closure space in both crisp and fuzzy environment, in addition, a characterization of the continuous mappings in quasi-discrete closure spaces is obtained. We propose the concept of quasi-discrete fuzzy closure spaces in the context of fuzzy sets and establish a one to one correspondence between quasi-discrete fuzzy closure spaces and reflexive fuzzy relations. We also discuss the relationship between topological closure c_T and closure c in quasi-discrete fuzzy closure spaces and show that the process from closure c to topological closure c_T can be realized via the process from a reflexive fuzzy relation to its transitive closure.