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Effective Borel degrees of some topological functions

✍ Scribed by Guido Gherardi

Publisher
John Wiley and Sons
Year
2006
Tongue
English
Weight
287 KB
Volume
52
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

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)

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