Incomputability of Simply Connected Planar Continua

TL;DR

This paper demonstrates the existence of certain simply connected planar continua with complex computability properties, answering a question about the presence of computable points in such sets.

Contribution

It constructs examples of co-c.e. planar continua with specific incomputability features, advancing understanding of computability in topological spaces.

Findings

Existence of a contractible planar co-c.e. dendroid without computable points

Examples of co-c.e. and computable dendrites with specific non-inclusion properties

Counterexamples to the inclusion of certain computable and co-c.e. continua

Abstract

Le Roux and Ziegler asked whether every simply connected compact nonempty planar co-c.e. closed set always contains a computable point. In this paper, we solve the problem of le Roux and Ziegler by showing that there exists a contractible planar co-c.e. dendroid without computable points. We also provide several pathological examples of tree-like co-c.e. continua fulfilling certain global incomputability properties: there is a computable dendrite which does not *-include a co-c.e. tree; there is a co-c.e. dendrite which does not *-include a computable dendrite; there is a computable dendroid which does not *-include a co-c.e. dendrite. Here, a continuum A *-includes a member of a class P of continua if, for every positive real, A includes a P-continuum B such that the Hausdorff distance between A and B is smaller than the real.