Abstract

A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Kunzi in the survey paper Nonsymmetric Topology ([Kun93], problem 7 ). 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 #-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 #-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.

Citations