Incomputability of Simply Connected Planar Continua

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.

💡 Research Summary

The paper investigates the interplay between topological connectivity and computability for closed subsets of Euclidean space. The central question, posed by Le Rou​x and Ziegler, asks whether every non‑empty simply connected compact planar Π⁰₁ (co‑c.e.) set must contain a computable point. The authors answer this in the negative by constructing a contractible planar dendroid that is Π⁰₁ yet contains no computable points.

The construction proceeds in several stages. First, a “basic dendrite” B is defined as a countable union of vertical line segments whose tops lie on the line y = 0. Using an incomputable computably enumerable (c.e.) set A ⊂ ℕ, the authors modify each “rising” of B: for each t ∈ ℕ they assign a width w(t) that is positive exactly when t ∈ A and zero otherwise. The resulting set D is still a dendrite, and because the sequence {w(t)} is uniformly computable, D is a computable closed set. However, the authors prove that D cannot *‑include any Π⁰₁ tree: if a Π⁰₁ subtree T ⊂ D existed, the set B = {t | \hat B_t ∩ T = ∅} would be c.e., which would make the complement ℕ \ A c.e., contradicting the incomputability of A. Hence D is a computable dendrite that is not almost computable.

Next, the paper introduces an embedding Ψ that maps any binary tree T ⊂ 2^{<ℕ} into the plane by connecting the points ψ(σ) for strings σ in T with line segments. Lemma 7 shows that T is Π⁰₁ (or c.e., or computable) iff Ψ(T) is a Π⁰₁ (or c.e., or computable) dendrite. Using this correspondence, the authors produce three further pathological examples:

1. A Π⁰₁ dendrite that does not *‑include any computable dendrite. This is obtained by taking a Π⁰₁ tree T_A coding the incomputable set A and considering Ψ(T_A).

2. A computable dendrite that does not *‑include any Π⁰₁ dendrite. Here a computable tree S is chosen, and Ψ(S) yields the desired dendrite.

3. A computable dendroid that does not *‑include any Π⁰₁ dendrite, demonstrating that the *‑inclusion relation is not symmetric even between dendroids and dendrites.

Finally, by carefully arranging the “rising” widths and ensuring that all branching points are isolated, the authors construct a planar dendroid that is contractible (hence simply connected) and Π⁰₁, yet contains no computable points. This provides a concrete counterexample to the Le Rou​x‑Ziegler question.

The paper also discusses “almost computability”: while every Π⁰₁ chainable continuum is almost computable (Iljazović), the computable dendrite D constructed above fails this property, showing that almost computability does not extend to all dendrites. Moreover, the authors distinguish between effective pathwise connectivity (which D possesses) and semi‑effective arcwise connectivity (which D lacks), illustrating subtle differences between classical topological notions and their effective counterparts.

In summary, the work demonstrates that strong topological constraints such as simple connectivity or contractibility do not guarantee the existence of computable points in Π⁰₁ planar sets. The *‑inclusion framework reveals a rich hierarchy of global incomputability phenomena among dendrites, dendroids, and trees, and the results have significant implications for computable analysis, effective topology, and the theory of non‑basis theorems.