Some mathematical structures of generalized rough sets in infinite universes of discourse

This paper presents a general framework for the study of mathematical structure of rough sets in infinite universes of discourse. Lower and upper approximations of a crisp set with respect to an infinite approximation space are first defined. Properties of rough approximation operators induced from various approximation spaces are examined. The relationship between a topological space and rough approximation operators is further established. By the axiomatic approach, various classes of rough approximation operators are characterized by different sets of axioms. The axiom sets of rough approximation operators guarantee the existence of certain types of crisp relations producing the same operators. The measurability structures of rough set algebras are also investigated. Finally, the connections between rough sets and Dempster-Shafer theory of evidence are also explored.