Hyperprojective hierarchy of qcb 0 -spaces

We extend the recently introduced Luzin hierarchy of qcb₀-spaces to all countable ordinals, obtaining in this way the hyperprojective hierarchy of qcb₀-spaces. We generalize all main results for the former hierarchy to this larger hierarchy. In particular, we extend the Kleene–Kreisel continuous functionals of finite types to the continuous functionals of countable types and relate them to the new hierarchy. We show that the category of hyperprojective qcb₀-spaces has much better closure properties than the category of projective qcb₀-spaces. As a result, there are natural examples of spaces that are hyperprojective but not projective.