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

TL;DR

This work determines the exact descriptive complexity of the set of arc-connected compact subsets of the plane. It develops a detailed arc-lifting and coding framework, including oriented-arc composition, uniform liftings, and triod-trap constructions, specialized to planar topology. The main result shows that the planar arc-connected hyperspace ${\mathcal C}_{\rm arc}(\mathbb{R}^2)$ lies in the class $\\check{\mathcal A}(\\boldsymbol\Pi^1_1)$ and is complete for this class, contrasting with the higher (but not reach) complexity $\\boldsymbol\Pi^1_2$ for arc-connected subsets in $\mathbb{R}^3$. The results yield a tight plane-specific bound and provide coding tools that tie hyperspace descriptive set theory to planar topology, with implications for uniform arc-liftings and the structure of arc-components in planar Polish spaces.

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 $\boldmath Σ^1_1$ and $\boldmath Π^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath Π^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$.

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Theorems & Definitions (93)

• Proposition 1.1
• Definition 1.2
• Proposition 1.3
• Theorem 1.4
• proof
• Theorem 1.6
• Lemma 1.7
• proof
• Lemma 1.8
• proof