The descriptive class of the set of all compact arc-connected subsets of \(\mathbb R^2\)
We compute the exact descriptive class of the set of all compact arc-connected subsets of \(\mathbb R^2\), which turns out to be strictly higher than the classical \(\boldsymbol\Sigma_1^1\) and \(\Pi_1^1\) classes of analytic and coanalytic sets, but strictly lower than the second level projective class \(\Pi_2^1\) which is the exact descriptive class of the set of all compact arc-connected subsets of \(\mathbb R^3\). This computation relies on some structural results concerning two-dimensional arc-connected sets.Debs, Saint Raymond: On the arc-wise connection relation in the plane
Debs, Saint Raymond: The descriptive complexity of the set of arc-connected compact subsets of the plane