Continuity Spaces: Reconciling Domains and Metric Spaces

We use continuity spaces, a common refinement of posets and metric spaces, to develop a general theory of semantic domains which includes metric spaces and domains of cpo’s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming. Our proposal for a general notion of semantic domain is a symmetrically compact Y-continuity space, where V is a value quantale. We show that the category of symmetrically compact V-continuity spaces with continuous maps has many of the key properties required of a category of domains and that it captures, in a natural way, the traditional examples. In general, the category will not be Cartesian closed; however, powerdomains do exist and, by adapting a construction of Suenderhauf to continuity spaces, we show that they define a computational monad in the sense of Moggi.