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| Titel: |
Incomputability of Simply Connected Planar Continua |
| Auteur: |
Kihara, Takayuki |
| Verschenen in: |
Computability |
| Paginering: | Jaargang 1 (2013) nr. 2 pagina's 131-152 |
| Jaar: |
2013-01-08 |
| Inhoud: |
Le Roux and Ziegler asked whether every simply connected compact nonempty planar $\Pi^0_1$ set always contains a computable point. In this paper, we solve the problem of le Roux and Ziegler by showing that there exists a planar $\Pi^0_1$ dendroid without computable points. We also provide several pathological examples of tree-like $\Pi^0_1$ continua fulfilling certain global incomputability properties: there is a computable dendrite which does not *-include a $\Pi^0_1$ tree; there is a $\Pi^0_1$ dendrite which does not *-include a computable dendrite; there is a computable dendroid which does not *-include a $\Pi^0_1$ dendrite. Here, a continuum A *-includes a member of a class $\mathcal{P}$ of continua if, for every positive real ε, A includes a continuum $B \in \mathcal{P}$ such that the Hausdorff distance between A and B is smaller than ε. |
| Uitgever: |
IOS Press |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
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