Valuations are measure-like functions mapping the open sets of a topological space into positive real numbers. They can be classified according to some additional properties. Some topological spaces are defined whose elements are valuations from various classes. The relationships among these spaces are studied, and universal properties are shown for some of them. 1 Introduction For a topological space X , a valuation on X is a function which maps the open sets of X to real numbers in the range from zero to infinity (inclusively) with the following properties: (1) The empty set is mapped to zero: ; = 0 (strictness). (2) The values assigned to binary union and intersection are related by the following equation: (U [ V ) + (U " V ) = U + V for all opens U and V (modularity). (3) Bigger sets are mapped to bigger numbers: if U ` V , then U V (monotonicity). Most often, we consider Scott continuous valuations which enjoy the additional property ( S i2I V i ) = t i2I V i for every dir...

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.

This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6. 0 Introduction In this section, we provide a common framework for the summarized papers. In Subsection 0.1, Moggi's approach to specify denotational semantics by means of strong monads is introduced. In Subsection 0.2, we specialize this approach to languages with a binary choice construct. Strong monads can be obtained in at least two ways: as free constructions w.r.t. algebraic theories (Subsection 0.3), and by using second order functions (Subsection 0.4). Finally, formal definitions of those concepts which are used in all...