Formal Topology and Domains

Formal topology and domains

The theme of this paper is the relation between formal topology and the theory of domains. On one hand, domain theory can be seen as a branch of formal topology. On the other hand, the influence of domain theory on formal topology is twofold. Historically, the presence of the subset Con in Scott’s information systems has been the starting point for the introduction of the positivity predicate P...

Apartness and formal topology

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 ...

General Recursion and Formal Topology

It is well known that general recursion cannot be expressed within Martin-Löf’s type theory and various approaches have been proposed to overcome this problem still maintaining the termination of the computation of the typable terms. In this work we propose a new approach to this problem based on the use of inductively generated formal topologies.

Sublocales in formal topology

The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each...

Formal Topology with Posets