In this chapter we investigate some ways domains can be used in order to study other mathematical structures of interest. In Sections 8.1 and 8.2 we consider ultrametric spaces and algebras and show how these can be topologically embedded into domains in a simple way. In Section 8.3 we consider the problem of abstracting the “total” elements of a domain from the partial ones. Then, in Section 8.4, this is used to show how the Kleene–Kreisel continuous functionals are represented in the partial continuous functionals.

Section 8.1 Metric and Ultrametric Spaces

In a general topological space, open sets are used to separate points or sets of points from each other. In many concrete spaces of interest one can do much more in that there is a natural function or metric which to each pair of points assigns a distance between the points, a non-negative real number. It was Fréchet [1906] who abstracted the properties needed from natural metrics on concrete spaces, such as the Euclidean spaces, in order to develop an abstract theory of metric spaces. Chronologically, the theory of metric spaces preceded the general theory of topological spaces introduced by Hausdorff, so that the abstraction to metric spaces was a first important step in obtaining general topological spaces. On the other hand, the metric spaces form an important subclass of the topological spaces.

Metric spaces have been used with success by several authors in order to give semantics for certain programming language constructs.