A study of rough sets based on 1-neighborhood systems

A special class of neighborhood systems, called 1-neighborhood systems, are important in rough set theory. By using a concept “core” originated in general topology, we define two types of rough sets based on 1-neighborhood systems in this paper. We discuss properties of these rough sets from the perspective of both common 1-neighborhood systems and several special classes of 1-neighborhood systems, such as reflexive, symmetric, transitive, or Euclidean 1-neighborhood systems. By using these properties, we discuss the relationship among several classes of 1-neighborhood systems with various special properties. We give a necessary and sufficient condition for a reflexive and symmetric 1-neighborhood system being a unary. We also prove that a reflexive and transitive 1-neighborhood system is representative. The proofs of these results show that the rough sets we defined in this paper not only have application background, but also have theoretic importance.