Affiliations: Kurt Gödel Research Center, University of Vienna, Vienna, Austria. E-mail: [email protected] | A.P. Ershov Institute of Informatics Systems SB RAS, Novosibirsk, Russia. E-mail: [email protected]
Abstract: 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.