The Continuous Functionals of Finite Types Over the Reals

Conference paper
Part of the Semantic Structures in Computation book series (SECO, volume 1)


We investigate a hierarchy of domains with totality where we close some selected base domains, including domains for the reals, the natural numbers and the boolean values, under cartesian products and restricted function spaces. We show that the total objects will be dense in the respective domains, and that our construction is equivalent to the analogue construction in the category of limit spaces.

In order to obtain this we will consider a restricted function space construction. We then show that this restriction, up to equivalence, does not restrict the class of total objects.


Algebraic domains reals finite types total functionals density topology limit spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauer, A., Birkedal, L. and Scott, D.S. Equilogical spaces, Manuscript (1998)Google Scholar
  2. 2.
    Berger, U. Totale Objecte and Mengen in der Bereichtheorie (in German), Thesis, München 1990.Google Scholar
  3. 3.
    Berger, U. Total sets and objects in domain theory, Annals of Pure and Applied Logic 60 (1993) 91-117.CrossRefGoogle Scholar
  4. 4.
    Ershov, Y.L. Model C of partial continuous functionals, in R.O. Gandy and J.M.E. Hyland (eds.) Logic Colloquium ’76, North Holland (1977) 455-467.Google Scholar
  5. 5.
    Hyland, J.M.E. Filter Spaces and Continuous Functionals, Annals of Mathematical Logic 16, (1979) 101-143.CrossRefGoogle Scholar
  6. 6.
    Kleene, S.C. Countable functionals, in A. Heyting (ed.): Constructivity in Mathematics, North-Holland (1959) 81-100Google Scholar
  7. 7.
    Kreisel, G. Interpretation of analysis by means of functionals of finite type, in A. heyting (ed.): Constructivity in Mathematics, North-Holland (1959) 101-128.Google Scholar
  8. 8.
    Kuratowski, C. Topologie Vol 1, Warsawa (1952).Google Scholar
  9. 9.
    Longo, G. and Moggi, E. The hereditarily partial effective functionals and recursion theory in higher types,Journal of Symbolic Logic 49, (1984) 1319-1332.CrossRefGoogle Scholar
  10. 10.
    Menni, M. and Simpson, A. The Largest Common Subcategory of Equilogical and Limit Spaces, manuscript (1998).Google Scholar
  11. 11.
    Normann, D. Recursion on the countable functionals, Springer Lecture Notes in Mathematics 811 (1980)Google Scholar
  12. 12.
    Normann,D.Categories of domains with totality Oslo preprint in Mathematics No 4 (1997). Revised version available on
  13. 13.
    Rummelhoff, I. Normal domain representation of topological spaces, To appear in Mathematical Logic Quarterly.Google Scholar
  14. 14.
    Scarpellini, B. A Model for Bar Recursion of Higher Types, Comp. Math. 23, (1971) 123-153.Google Scholar
  15. 15.
    Stoltenberg-Hansen, V., Lindström, I. and Griffor, E.R. Mathematical Theory of Domains,Cambridge University Press (1994).Google Scholar
  16. 16.
    Waagbø, G.A. Domains-with-totality semantics for Intuitionistic Type Theory, Thesis, Oslo 1997Google Scholar
  17. 17.
    Waagbø, G.A. Lifting Theorems in the theory of domains with totality,in preparation.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  1. 1.Department of MathematicsThe University of OsloOsloNorway

Personalised recommendations