Limit spaces with approximations

Abstracting from a presentation of the density theorem for the hierarchy Ct(ρ)Ct(ρ) of countable functionals over NN given by Normann in [12], we define two subcategories of limit spaces, the limit spaces with approximations, and the limit spaces with general approximations, for both of which a density theorem holds directly. We show that these categories are cartesian closed, and we give examples of such limit spaces and of density theorems for hierarchies of functionals over them. Most of our main proofs are within Bishop's informal system of constructive mathematics BISH. In a limit space with (general) approximations the approximation functions are given beforehand as an internal part of the structure under study. In this way limit spaces with (general) approximations form a constructive approach to abstract limit spaces, reflecting at the same time the central idea of Normann's Program of Internal Computability.