Domain theory -- how do partial metic spaces relate to Scott's theory of domains?

The conception of reconciling domains with metric spaces begins with work by Mike Smyth [Sm88]. The essential idea is that by dropping the symmetry axiom for metric spaces, which by definition is a quasi-metric space, we allow a reconciliation to be made with the inherent non symmetry of domain theory.

Using enriched categories Kim Wagner has established a general framework to unify the partial order and metric apporoaches to solving recursive domain equations [Wag94, Wag97].

Allusions with domain theory, via quasi-metric spaces, were made by Matthews [Mat94]. A weighted quasi-metric space is a quasi-metric space (X,q) with a so-called weight function | |:X->R⁺ such that q(x,y)+|x|=q(y,x)+|y|. Equivalent to partial metric spaces, these weighted quasi-metric spaces were subsequently studied in more detail by Hans-Peter Künzi and Václav Vajner [KV94]. Not every quasi-metric space is so weightable [Mat94], and so this raises the following interesting question. Which domains are partially metrizable.

In his PhD thesis [Was02] Pawel Waszkiewicz made extensive studies of the relationship between partial metric spaces and domains. He has characetrized (completely) those algebraic domains which are partially metrizable [Was03a]. Waszkiewicz has given many examples of partial metrics for domains, such as a canonical partial metric for Plotkin's T-omega universal domaini [Was03b]. This paper also contains other partial metrizability results, especially of interest are those partial metric spaces in which p(x,x)=0 if and only if x is maximal. Waszkiewicz has shown that every omega-continuous dcpo is partially metrizable, something that Michel Schellekens made a central point of in a later paper [Sch03a].

Kopperman et.al. [KMP04] have generalised partial metric spaces by generalising their range of the non negative reals to a value quantale. They show that any topology can be described by such a generalised partial metric space, and for each continuous poset, there is such a generalized metric whose topology is the Scott topology, and whose dual topology is the lower topology.