The descriptive complexity of the set of arc-connected compact subsets of the plane

Gabriel Debs; Jean Saint Raymond

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arXiv · 2601.13135

The descriptive complexity of the set of arc-connected compact subsets of the plane

Gabriel Debs·Jean Saint Raymond

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Abstract

We compute the exact complexity of the set of all arc-connected compact subsets of $\boldmath R^2$, which turns out to be strictly higher than the classical $\boldmath \Sigma^1_1$ and $\boldmath \Pi^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath \Pi^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$.

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Gabriel Debs, Jean Saint Raymond. 2026-01-19. The descriptive complexity of the set of arc-connected compact subsets of the plane. https://arxiv.org/abs/2601.13135

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