Neighborhood systems and covering approximation spaces

The approximation theory is studied via a 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 the concept 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. We regard a given covering as a special form of a total pure reflexive neighborhood system, called a covering neighborhood system, by assigning to each object x the nonempty family CN(x) of all members of the covering that contain x. We then investigate the approximation structure of the covering by introducing two pairs of lower and upper approximations: one pair treats each CN(x) as a local base, the other pair as a local subbase. We obtain optimal lower and upper approximations of a covering of an approximation space.