On Topologies Defined by Binary Relations in Rough Sets

In the theory of rough sets of data-mining, a subset of a database represents a certain knowledge. Thus to determine the subset in the database is equivalent to obtain the knowledges which the database possesses. A topological space is constructed by the database. An open subset in the topological space defined by the database corresponds to a certain knowledge in the database. Here we consider topological properties of approximation spaces in generalized rough sets. We show that (a) If R is reflexive and transitive, then R = R (T(R)). Conversely, if R = R(T (R)), then R is reflexive and transitive.(b)If O is a topology with a property (IP), then O = T(R(O)), where (IP) means that Aλ ∈ O(λ ∈ Λ) implies ∩λ Aλ ∈ O. Conversely, for any topology O, if O=T(R(O)), then it satisfies (IP).