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Theory of Computing Systems

, Volume 60, Issue 1, pp 53–111 | Cite as

Notes on Computable Analysis

  • Michelle Porter
  • Adam Day
  • Rodney Downey
Article
Part of the following topical collections:
  1. 50th Anniversary

Abstract

Computable analysis has been part of computability theory since Turing’s original paper on the subject (Turing, Proc. London Math. Sc. 42:230–265, 1936). Nevertheless, it is difficult to locate basic results in this area. A first goal of this paper is to give some new simple proofs of fundamental classical results (highlighting the role of \({{\Pi }_{1}^{0}}\) classes). Naturally this paper cannot cover all aspects of computable analysis, but we hope that this gives the reader a completely self-contained ingress into this area. A second goal is to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpiński, and the Blaschke Selection Theorem.

Keywords

Computable analysis Computable real Computable real-valued function Markov computable Borel computable Type II computable Darboux property Blaschke Selection Theorem Convex sets 

Notes

Acknowledgments

This work was supported by the Marsden Fund of New Zealand, and a MSc scholarship to Porter. Much of the material is based around Porter’s MSc Thesis supervised by Day and Downey, and for this reason Porter is the first author on the paper.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsVictoria UniversityWellingtonNew Zealand

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