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Effective Borel degrees of some topological functions
Author(s) -
Gherardi Guido
Publication year - 2006
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200610021
Subject(s) - mathematics , borel hierarchy , closure (psychology) , topological space , completeness (order theory) , borel set , intersection (aeronautics) , metric space , discrete mathematics , pure mathematics , borel measure , probability measure , mathematical analysis , economics , engineering , market economy , aerospace engineering
The focus of this paper is the incomputability of some topological functions (with respect to certain representations) using the tools of Borel computability theory, as introduced by V. Brattka in [3] and [4]. First, we analyze some basic topological functions on closed subsets of ℝ n , like closure, border, intersection, and derivative, and we prove for such functions results of Σ 0 2 ‐completeness and Σ 0 3 ‐completeness in the effective Borel hierarchy. Then, following [13], we re‐consider two well‐known topological results: the lemmas of Urysohn and Urysohn‐Tietze for generic metric spaces (for the latter we refer to the proof given by Dieudonné). Both lemmas define Σ 0 2 ‐computable functions which in some cases are even Σ 0 2 ‐complete. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)