UPPER AND LOWER POWERSPACES OF DIRECTED SPACES

In domain theory, powerdomains play an important role in modeling the semantics of nondeterministic functional programming languages. We extend the notion of powerdomains to the cartesian closed category of directed spaces. We define the notion of upper and lower powerspace of a directed space by the way of free algebras. We show that the upper and lower powerspaces of any directed space exist and give their concrete structures. Moreover, the upper and lower functors preserve the continuity of directed spaces. The upper and lower powerspaces of c-spaces (resp. FS-spaces) are c-spaces (resp. FS-spaces).