The descriptive complexity of the set of arc-connected compact subsets of the plane
Volume 273 / 2026
Fundamenta Mathematicae 273 (2026), 217-250
MSC: Primary 54H05; Secondary 03E15
DOI: 10.4064/fm241114-10-4
Published online: 8 June 2026
Abstract
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 $\boldsymbol\Pi^1_1$ classes of analytic and coanalytic sets, but strictly lower than the class $\boldsymbol\Pi ^1_2$ which is the exact descriptive class of the set of all compact arc-connected subsets of $\mathbb R^3$.
