An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability

  • Guido Gherardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

In [6], K. Weihrauch studied the computational properties of the Urysohn Lemma and of the Urysohn-Tietze Lemma within the framework of the TTE-theory of computation. He proved that with respect to negative information both lemmas cannot in general define computable single valued mappings. In this paper we reconsider the same problem with respect to positive information. We show that in the case of positive information neither the Urysohn Lemma nor the Dieudonné version of Urysohn-Tietze Lemma define computable functions. We analyze the degree of the incomputability of such functions (or more precisely, of the incomputability of some of their realizations in the Baire space) according to the theory of effective Borel measurability. In particular, we show that with respect to positive information both the Urysohn function and the Dieudonné function are \(\Sigma^{\rm 0}_{\rm 2}\)-computable and in some cases even \(\Sigma^{\rm 0}_{\rm 2}\)-complete.

Keywords

Computable Analysis Borel Measurability Urysohn Lemma Urysohn-Tietze Lemma 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brattka, V.: Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51, 19–44 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brattka, V.: On the Borel complexity of Hahn-Banach extensions. Electronic Notes in Theoretical Computer Science 120, 3–16 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theoretical Computer Science 305, 43–76 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1960)MATHGoogle Scholar
  5. 5.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  6. 6.
    Weihrauch, K.: On computable metric spaces Tietze-Urysohn extension is computable. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 357–368. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Guido Gherardi
    • 1
  1. 1.Dipartimento di Scienze Matematiche e Informatiche“R. Magari”, Università di SienaSienaItaly

Personalised recommendations