On covering generalized rough set: Covering rough membership function

Original rough membership function is defined using equivalence classes, then it is extended to topological spaces. Note that both equivalence class and topological space form some covering on a universe of discourse. In this paper, the notion of rough membership function is generalized to arbitrary coverings. The properties of this conception are studied, and an axiomatical definition of that is presented. In addition, it is proved that this notion integrates the concept of covering rough set and fuzzy sets. To our knowledge, the axiomatic description for rough membership function is stated for the first time.