Abstract
In this chapter we lay the basis for a unified Type 2 recursion theory. Our investigations from Chapter 3.1 and especially Theorem 3.1.14 show that it suffices to study continuity and computability on Baire’s space B . We shall choose the set B = ℕℕ as our standard set on which we develop a theory of continuity (w.r.t. Baire’s topology) and of computability. The theory turns out to be formally similar to Type 1 recursion theory. Most definitions and theorems will appear in a topological and in a computational version in accordance with our previously explained program for Type 2 theory. First we introduce a representation \( \hat \varphi \) : B →[B → Bo] of the continuous functions from B to Bo which is effective: it satisfies the Type 2 utm-theorem and smn-theorem. From \( \hat \varphi \) we derive a representation φ : B →[B → B] where [B →B] is the set of continuous functions г: B → B the domains of which are the Gδ-subsets of B , and we derive a representation x: B→[B→ℕ] of the set of the continuous functions г: B → ℕ the domains of which are the open subsets of B . The representations φ and x satisfy the utm-theorem and the smn-theorem. As a counterpart of the recursively enumerable subsets in Type 1 theory we study the open and the computably open subsets of B. This Type 2 theory of open subsets of B is formally very similar to the Type 1 theory of the recursively enumerable sets.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Weihrauch, K. (1987). Recursion Theory on Baire’s Space. In: Computability. EATCS Monographs on Theoretical Computer Science, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-69965-8_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-69965-8_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-69967-2
Online ISBN: 978-3-642-69965-8
eBook Packages: Springer Book Archive