M. Kondo (Japan)

approximation space, binary relation, topology

In the theory of rough sets of data-mining, a subset of a database represents a certain knowledge. Thus to deter mine 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 topo logical 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).

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