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)


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.


Computable Analysis Borel Measurability Urysohn Lemma Urysohn-Tietze Lemma 


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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

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

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