On the Structure of the Dual Complexity Space: the General Case

The purpose of this note is to report the main results obtained by the authors in [8] and [9], respectively. The notion of a Smyth completable quasi-uniform space provides an efficient tool to give a topological foundation for many kinds of spaces which arise naturally in Theoretical Computer Science; in particular in Domain Theory (e.g. [5], [13] and [14]) and Complexity Theory (e.g. [9] and [10]). In fact, Smyth presented in [13] and [14] a topological framework for denotational semantic based on the theory of complete (and totally bounded) quasi-uniform and quasi-metric spaces. Later on, Matthews introduced in [5] the weightable quasi-metric spaces, or the equivalent partial metric spaces, as a part of the study of denotational semantics of dataflow networks. It was proved by Künzi [4] that, in fact, every weigthable quasi-metric space is Smyth completable. Recently, Schellekens [10] introduced the complexity (quasi-metric) space to the study of complexity analysis of programs and proved, among other results, that every complexity space is weightable and, thus, Smyth completable. Our basic references for quasi-metric spaces are [3] and [4]. In our context by a quasi-metric on a (nonempty) set X we mean a nonnegative real-valued function d on X×X such that for all x, y, z ∈ X : (i) d(x, y) = d(y, x) = 0⇔ x = y, and (ii) d(x, y) ≤ d(x, z) + d(z, y). If d is a quasi-metric on X, then the function d defined on X × X by d(x, y) = max{d(x, y), d(y, x)} is a metric on X. ∗The first author acknowledges the support of the DGES, grant PB95-0737.