Admissible domain representations of inductive limit spaces

In this paper we consider the following separate but related questions: “How do we construct an effective admissible domain representation of a space X from a pseudobasis on X?”, and “How do we construct an effective admissible domain representation of the inductive limit of a directed system (Xi)i∈I of topological spaces, given effective admissible domain representations of the individual spaces Xi?”. An attempt to answer the first of these questions leads us naturally to the notion of an admissible pair. To answer the second question we employ admissible pairs to characterise a subclass of admissible domain representations which is closed under inductive limits. The second result is particularly interesting from the point of view of computable analysis, since it allows us to construct an effective and admissible domain representation of the space of test functions which is of interest in distribution theory.