Hyperprojective Hierarchy of qcb0-Spaces
We extend the Luzin hierarchy of qcb0-spaces introduced in [ScS13] to all countable ordinals, obtaining in this way the hyperprojective hierarchy of qcb0-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 qcb0-spaces has much better closure properties than the category of projective qcb0-space. As a result, there are natural examples of spaces that are hyperprojective but not projective.
KeywordsHyperprojective hierarchy qcb0-space continuous functionals of countable types cartesian closed category
Unable to display preview. Download preview PDF.
- [Ke83]Kechris, A.S.: Suslin cardinals, k-Suslin sets and the scale property in the hyperprojective hierarchy. In: Kechris, A.S., Löwe, B., Steel, J.R. (eds.) The Cabal Seminar, v. 1: Games, Scales and Suslin Cardinals. Lecture Notes in Logic, vol. 31, pp. 314–332 (2008); (reprinted from Lecture Notes in Mathematica, No 1019. Springer, Berlin 1983)Google Scholar
- [Kl59]Kleene, S.C.: Countable functionals. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 87–100. North Holland, Amsterdam (1959)Google Scholar
- [Kr59]Kreisel, G.: Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 101–128. North Holland, Amsterdam (1959)Google Scholar
- [Sch03]Schröder, M.: Admissible representations for continuous computations. PhD thesis, Fachbereich Informatik, FernUniversität Hagen (2003)Google Scholar
- [ScS13]Schröder, M., Selivanov, V.: Some Hierarchies of qcb0-Spaces. Mathematical Structures in Computer Science, arXiv:1304.1647 (to appear)Google Scholar