The Continuous Functionals of Finite Types Over the Reals
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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.
KeywordsAlgebraic domains reals finite types total functionals density topology limit spaces
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- 1.Bauer, A., Birkedal, L. and Scott, D.S. Equilogical spaces, Manuscript (1998)Google Scholar
- 2.Berger, U. Totale Objecte and Mengen in der Bereichtheorie (in German), Thesis, München 1990.Google Scholar
- 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
- 6.Kleene, S.C. Countable functionals, in A. Heyting (ed.): Constructivity in Mathematics, North-Holland (1959) 81-100Google Scholar
- 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.Kuratowski, C. Topologie Vol 1, Warsawa (1952).Google Scholar
- 10.Menni, M. and Simpson, A. The Largest Common Subcategory of Equilogical and Limit Spaces, manuscript (1998).Google Scholar
- 11.Normann, D. Recursion on the countable functionals, Springer Lecture Notes in Mathematics 811 (1980)Google Scholar
- 12.Normann,D.Categories of domains with totality Oslo preprint in Mathematics No 4 (1997). Revised version available on http://www.math.uio.no/dnormann/Categori es.ps. http://www.math.uio.no/dnormann/Categori/es.ps
- 13.Rummelhoff, I. Normal domain representation of topological spaces, To appear in Mathematical Logic Quarterly.Google Scholar
- 14.Scarpellini, B. A Model for Bar Recursion of Higher Types, Comp. Math. 23, (1971) 123-153.Google Scholar
- 15.Stoltenberg-Hansen, V., Lindström, I. and Griffor, E.R. Mathematical Theory of Domains,Cambridge University Press (1994).Google Scholar
- 16.Waagbø, G.A. Domains-with-totality semantics for Intuitionistic Type Theory, Thesis, Oslo 1997Google Scholar
- 17.Waagbø, G.A. Lifting Theorems in the theory of domains with totality,in preparation.Google Scholar