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. (C) 2013 Elsevier Inc. All rights reserved.