A characterization of partial metrizability: domains are quantifiable

A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Künzi in the survey paper Nonsymmetric Topology ([Kün93], problem 71). The characterization yields a powerful tool which establishes a correspondence between partial metrics and special types of valuations, referred to as Q-valuations (cf. also [Sch00]). The notion of a Q-valuation essentially combines the well-known notion of a valuation with a weaker version of the notion of a quasi-unimorphism, i.e. an isomorphism in the context of quasi-uniform spaces. As an application, we show that ω-continuous dcpo’s are quantifiable in the sense of [O’N97], i.e. the Scott topology and partial order are induced by a partial metric. For ω-algebraic dcpo’s the Lawson topology is induced by the associated metric. The partial metrization of general domains improves prior approaches in two ways: The partial metric is guaranteed to capture the Scott topology as opposed to e.g. [Smy87],[BvBR95],[FS96] and [FK97], which in general yield a coarser topology. Partial metric spaces are Smyth-completable and hence their Smyth-completion reduces to the standard bicompletion. This type of simplification is advocated in [Smy91]. Our results extend [Smy91]’s scope of application from the context of 2/3 SFP domains to general domains. The quantification of general domains solves an open problem on the partial metrizability of domains2 stated in [O’N97] and [Hec96]. Our proof of the quantifiability of domains is novel in that it relies on the central notion of a semivaluation ([Sch00]). The characterization of partial metrizability is entirely new and sheds light on the deeper connections between partial metrics and valuations commented on in [BSh98]. Based on [Sch00] and our present characterization, we conclude that the notion of a (semi)valuation is central in the context of Quantitative Domain Theory since it can be shown to underlie the various models arising in the applications. AMS Subject Classification: 54E15, 54E35, 06A12, 06B35