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) 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) 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.
neighborhood systems covering approximation spaces local base local subbase topological neighborhood axioms rough sets