The standard Pawlak approach to rough set theory, as an approximation space consisting of a universe U and an equivalence (indiscernibility) relation R U x U, can be equivalently described by the induced preclusivity ("discernibility") relation U x U \ R, which is irreflexive and symmetric.We generalize the notion of approximation space as a pair consisting of a universe U and a discernibility or preclusivity (irreflexive and symmetric) relation, not necessarily induced from an equivalence relation. In this case the "elementary" (...) sets are not mutually disjoint, but all the theory of generalized rough sets can be developed in analogy with the standard Pawlak approach. On the power set of the universe, the algebraic structure of the quasi fuzzy-intuitionistic "classical" (BZ) lattice is introduced and the sets of all "closed" and of all "open" definable sets with the associated complete (in general nondistributive) ortholattice structures are singled out. (shrink)