Comparing hierarchies of total functionals

In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional” representation of the reals and the other on an “intensional” representation. The problem is if these two hierarchies coincide. We will show that this coincidence problem is equivalent to the statement that the topology on the Kleene-Kreisel continuous functionals of a fixed type induced by all continuous functions into the reals is zero-dimensional for each type. As a tool of independent interest, we will construct topological embeddings of the Kleene-Kreisel functionals into both the extensional and the intensional hierarchy at each type. The embeddings will be hierarchy embeddings as well in the sense that they are the inclusion maps at type 0 and respect application at higher types.