Representations of FS-domains and BF-domains via FS-approximation Spaces

In this paper, concepts of (topological) FS-approximation spaces are introduced. Representations of FS-domains and BF-domains via (topological) FS-approximation spaces are considered. It is proved that the collection of CF-closed sets in an FS-approximation space (resp., a topological FS-approximation space) endowed with the set-inclusion order is an FS-domain (resp., a BF-domain) and that every FS-domain (resp., BF-domain) is order isomorphic to the collection of CF-closed sets of some FS-approximation space (resp., topological FS-approximation space) endowed with the set-inclusion order. The concept of topological BF-approximation spaces is introduced and a skillful method without using CF-approximable relations to represent BF-domains is given. It is also proved that the category of FS-domains (resp., BF-domains) with Scott continuous maps as morphisms is equivalent to that of FS-approximation spaces (resp., topological FS-approximation spaces) with CF-approximable relations as morphisms.