Institute of Information Science, Academia Sinica

The approximation theory is studied via a common generalization of pretopological and topological neighborhood systems, called total pure reflexive neighborhood systems. In the framework of such neighborhood systems, the definition of pre-topological equivalence is formulated in terms of neighborhoods; we show that the so-called lower and upper approximations are pre-topological invariants and dual to each other, and we construct the corresponding generalized topological closure via the upper approximation.

In addition, to illustrate the applications in rough set theory, two generalizations (U, β) in cases of reflexive relations and coverings, of Pawlak approximation spaces are considered.