Functionals of Type 3 as Realisers of Classical Theorems in Analysis

We investigate the relative computational strength of combinations of four higher order functionals, the jump and hyperjump seen as functionals of type 2 and realisers for the compactness of Cantor space and the Lindelöf property of Baire space seen as functionals of type 3. We compare them with the closure operator for non-monotone inductive definitions of sets of integers, also seen as a functional of type 3