On the topological properties of generalized rough sets

In this paper, we consider some topological properties of generalized rough sets induced by binary relations and show that1.Any serial binary relation can induce a topology.2.Let R be a binary relation on a universe U. t(R) and e(R) denote the transitive closure and the equivalence closure of R, respectively. If R is a reflexive relation on U, then R and t(R) induce the same topology, i.e. T(R) = T(t(R)). The interior and closure operators of the topology T(R) induced by R are the lower and upper approximation operators t(R) and t(R)‾, respectively. Moreover, R(T(R)) = t(R), where R(T(R)) is the relation induced by the topology T(R).3.When R is a reflexive and symmetric relation, R and e(R) induce the same topology, i.e. T(R) = T(e(R)). The interior and closure operators of the topology T(R) induced by R are the lower and upper approximation operators e(R) and e(R)‾, respectively. Moreover, R(T(R)) = e(R).4.Based on the above conclusions, the notion of topological reduction of incomplete information systems is proposed, and characterizations of reduction of consistent incomplete decision tables are obtained.