A partial spatial object is a partial map from space to data. Data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry (CVG), where space and data are subspaces of # dimensional Euclidean space. We investigate the computability of partial spatial object data types, in general and in volume graphics, using the theory of effective domain representations for topological algebras. The basic mathematical problem considered is to classify which partial functions between topological spaces can be represented by total continuous functions between given domain representations of the spaces. We prove theorems about partial functions on regular Hausdorff spaces and their domain representations, and apply the results to partial spatial objects and CVG algebras.

This paper gives effective domain representations of spaces H(X) of non-empty compact subsets of effective complete metric spaces X. The domain representation of H(X) is constructed from a domain representation of X using the Plotkin power domain construction. As an application of the representation an effective version of a fundamental theorem on IFS (iterated function system) is shown.

In this paper we consider admissible domain representations of topological spaces. A domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to D via a continuous function from E to D. We present a characterisation theorem of when a topological space has a λ-admissible and κ-based domain representation. We also prove that there is a natural cartesian closed category of countably based and countably admissible domain representations. These results are generalisations of [Sch02]. 1

It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison d’être for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that the category of admissibly representable spaces with morphisms functions which are representable by a partial continuous function is Cartesian closed. Finally, we consider the question of effectivity. Key words. Domain theory, domain representations, computability theory, computable analysis. 1

It is shown that every compact metric space X is homeomorphically embedded in an !-algebraic domain D as the set of minimal limit elements.

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1.

on topological spaces via domain representations

The theory of formal spaces and the more recent theory of apartness spaces have a priori not much more in common than that each of them was initiated as a constructive approach to general topology. We nonetheless try to do the first steps in relating these competing theories to each other. Formal topology was put forward in the mid 1980s by Sambin [8] in order to make available to Martin–Löf’s type theory [7] the concepts of classical topology that are worth keeping to such a constructive and predicative framework. In the meantime formal topology has proved a fairly universal setting for doing topology in a point–free way. We refer to [9] for a recent and exhaustive survey of formal topology. The theory of apartness spaces was started by Bridges and Vîță [4] nearly twenty years later to reformulate set–theoretic topology as an extension of Bishop’s constructive analysis [2, 3]. The subsequent development of the theory of apartness spaces has also shed some light on its classical counterpart, the theory of proximity or nearness spaces. A comprehensive overview will be available soon [5]. In formal topology ‘basic neighbourhood ’ is a primitive concept, whereas ‘point ’ is a derived notion; as sets of basic neighbourhoods, points have to be handled with particular care to meet the needs of a predicative framework like Martin–Löf type theory. In the theory of apartness spaces, it is the other way round: as in classical topology, points are given as such, and (basic) neighbourhoods are sets of points. Since, however, it is hard to detect any truly impredicative move in the practice of Bishop’s constructive mathematics in general, we dare to undertake the following attempt to link formal topology and the theory of apartness spaces to each other. 1 Basic definitions We recall the standard definitions associated with formal topologies and morphisms between them (approximable mappings).

Abstract. General methods of investigating effectivity on regular Hausdorff (T3) spaces is considered. It is shown that there exists a functor from a category of T3 spaces into a category of domain representations. Using this functor one may look at the subcategory of effective domain representations to get an effectivity theory for T3 spaces. However, this approach seems to be beset by some problems. Instead, a new approach to introducing effectivity to T3 spaces is given. The construction uses effective retractions on effective Scott–Ershov domains. The benefit of the approach is that the numbering of the basis and the numbering of the elements are derived at once. 1

is the standard domain representation of a quotient space admissible?