Computable analysis: an introduction
Computable analysis: an introduction
On Computable Metric Spaces Tietze-Urysohn Extension Is Computable
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
On the Borel Complexity of Hahn-Banach Extensions
Electronic Notes in Theoretical Computer Science (ENTCS)
Singular Coverings and Non-Uniform Notions of Closed Set Computability
Electronic Notes in Theoretical Computer Science (ENTCS)
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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.