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

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.