A note on the relationships between generalized rough sets and topologies

Abstract

Quite recently, Wu and Liu (2020) [11] raised an open problem when they discussed the relationships between generalized rough sets and topologies. Said precisely, each binary relation generates a topology through the lower rough approximation operator, then for two binary relations on the same set, is there a sufficient and necessary condition such that the union of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. In this note, we give a positive answer to this problem and provide a union-join condition. Furthermore, note that the main reason that the expected identity hold not consists in the fact that the union of two topologies is not a topology, in general. Then replacing the union of two topologies with the supremum of them, we give a sufficient and necessary condition (supremum-join condition) such that the supremum of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. We verify that the supremum-join condition is much simpler than the union-join condition. At last, we give a characterization on discrete topology generated by a binary relation using the sober separation.