Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enriched-categorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the ffl-ball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.

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.

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Every generalized metric space can be isometrically embedded in a complete function space by means of a metric version of the categorical Yoneda embedding. This simple fact gives naturally rise to: 1. a topology for generalized metric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary metric spaces reduces to the ffl-ball topology; 2. a topology for algebraic generalized metric spaces generalizing both the Scott topology for algebraic complete partial orders and the ffl-ball topology for metric spaces. AMS subject classification (1991): 68Q10, 68Q55 Keywords: generalized metric, preorder, metric, Alexandroff topology, Scott topology, ffl-ball topology, Yoneda embedding 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and [BV96]). Parts of their theory have been develop...