Hyperprojective Hierarchy of QCB_0-spaces

Abstract

We extend the Luzin hierarchy of qcb${}_0$-spaces introduced in [ScS13] to all countable ordinals, obtaining in this way the hyperprojective hierarchy of qcb${}_0$-spaces. We generalize all main results of [ScS13] 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 qcb${}_0$-spaces has much better closure properties than the category of projective qcb${}_0$-space. As a result, there are natural examples of spaces that are hyperprojective but not projective.

Conference version to appear in LNCS