Generalizing realizability and Heyting models for constructive set theory

This article presents a generalisation of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalisation of the Boolean models for classical set theory which are a kind of forcing, while realizability is a decidedly constructive method that has first been develloped for number theory by Kleene and was later very fruitfully adapted to constructive set theory. In order to achieve the generalisation, a new kind of structure (applicative topologies) is introduced, which contains both elements of formal topology and applicative structures. The generalisation not only deepens the understanding of class models and leads to more efficiency in proofs about these kind of models, but also makes it possible to prove new results about the special cases which were not known before and to construct new models. Generalising Realizability and Heyting Models for Constructive Set Theory