2011/10/27 by Takayuki Kihara, Kihara, Takayuki
Computer Science · Mathematics · #03D78 #03F60 #54D05 #Advanced Topology and Set Theory #Cellular Automata and Applications #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO) #cs.LO #math.LO #msc:03D78 #msc:03F60 #msc:54D05
paper · pdf · doi:10.48550/arxiv.1110.6140
25 pages
arxiv created 2011/10/27 · openalex publication_date 2011/10/27 · arxiv updated 2011/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.