Abstract

The focus of this paper is the incomputability of some topological functions 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 ℝn, like closure, border, intersection, and derivative, and we prove for such functions results of Σ02-completeness and Σ03-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 . Both lemmas define Σ02-computable functions which in some cases are even Σ02-complete