The Rice-Shapiro theorem in Computable Topology

TL;DR

This paper explores conditions under which the Rice-Shapiro theorem applies in computable topology, providing criteria, counterexamples, and constructions to understand its scope and limitations.

Contribution

It establishes specific requirements for effectively enumerable topological spaces to satisfy the Rice-Shapiro theorem and introduces constructions generating such spaces from wn-families and computable trees.

Findings

Identifies conditions ensuring the Rice-Shapiro theorem holds in certain spaces.

Shows that relaxing these conditions leads to spaces where the theorem fails.

Provides explicit constructions and examples illustrating these properties.

Abstract

We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable topological spaces where the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable topological spaces with particular properties from wn--families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class.