Hyperprojective Hierarchy of qcb0-Spaces

  • Matthias Schröder
  • Victor Selivanov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8493)


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.


Hyperprojective hierarchy qcb0-space continuous functionals of countable types cartesian closed category 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Br13]
    de Brecht, M.: Quasi-Polish spaces. Annals of Pure and Applied Logic 164, 356–381 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. [En89]
    Engelking, R.: General Topology. Heldermann, Berlin (1989)zbMATHGoogle Scholar
  3. [ELS04]
    Escardó, M., Lawson, J., Simpson, A.: Comparing Cartesian closed Categories of Core Compactly Generated Spaces. Topology and its Applications 143, 105–145 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. [GH80]
    Giertz, G., Hoffmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: A compendium of Continuous Lattices. Springer, Berlin (1980)CrossRefGoogle Scholar
  5. [Hy79]
    Hyland, J.M.L.: Filter spaces and continuous functionals. Annals of Mathematical Logic 16, 101–143 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  6. [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
  7. [Ke95]
    Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  8. [Kl59]
    Kleene, S.C.: Countable functionals. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 87–100. North Holland, Amsterdam (1959)Google Scholar
  9. [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
  10. [KW85]
    Kreitz, C., Weihrauch, K.: Theory of representations. Theoretical Computer Science 38, 35–53 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  11. [No80]
    Normann, D.: Recursion on the Countable Functionals. Lecture Notes in Mathematics, vol. 811. Springer, Heidelberg (1980)zbMATHGoogle Scholar
  12. [No81]
    Normann, D.: Countable functionals and the projective hierarchy. Journal of Symbolic Logic 46(2), 209–215 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  13. [No99]
    Normann, D.: The continuous functionals. In: Griffor, E.R. (ed.) Handbook of Computability Theory, pp. 251–275. Elsevier, Amsterdam (1999)CrossRefGoogle Scholar
  14. [Sch02]
    Schröder, M.: Extended admissibility. Theoretical Computer Science 284, 519–538 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. [Sch03]
    Schröder, M.: Admissible representations for continuous computations. PhD thesis, Fachbereich Informatik, FernUniversität Hagen (2003)Google Scholar
  16. [Sch09]
    Schröder, M.: The sequential topology on \(\mathbb{N}^{\mathbb{N}^{\mathbb{N}}}\) is not regular. Mathematical Structures in Computer Science 19, 943–957 (2009)CrossRefzbMATHGoogle Scholar
  17. [ScS13]
    Schröder, M., Selivanov, V.: Some Hierarchies of qcb0-Spaces. Mathematical Structures in Computer Science, arXiv:1304.1647 (to appear)Google Scholar
  18. [Se06]
    Selivanov, V.L.: Towards a descriptive set theory for domain-like structures. Theoretical Computer Science 365, 258–282 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. [Se13]
    Selivanov, V.L.: Total representations. Logical Methods in Computer Science 9(2), 1–30 (2013), doi:10.2168/LMCS-9(2:5)2013MathSciNetGoogle Scholar
  20. [We00]
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Matthias Schröder
    • 1
  • Victor Selivanov
    • 2
  1. 1.Kurt Gödel Research Center, University of ViennaAustria
  2. 2.A.P. Ershov Institute of Informatics Systems SB RASNovosibirskRussia

Personalised recommendations