A nonsmooth path-connectedness relation in the real plane

TL;DR

The paper addresses the descriptive set-theoretic complexity of the path-connectedness relation on compact subsets of the plane. It constructs a concrete compact subset $H\subset \mathbb{R}^2$—built from a Knaster continuum–style arrangement of semicircles—that encodes a Cantor-type equivalence structure, and proves that the path-connectedness relation $\approx_H$ is Borel bireducible to the nonsmooth hyperfinite relation $E^*_0$ (which is itself Borel bireducible to $E_0$). The core contribution is the two-sided reduction: a continuous map $\psi$ showing $E^*_0 \leq_B \approx_H$, and a Borel map $\varphi$ showing $\approx_H \leq_B E^*_0$, yielding $\approx_H \sim_B E^*_0$. This provides a positive answer to Becker's question about the existence of a compact planar set with a non-smooth path-connectedness relation and clarifies the maximal Borel complexity attainable by such relations within $\mathcal{K}(\mathbb{R}^2)$.

Abstract

In this paper, we construct a compact subset of the real plane whose path-connectedness equivalence relation is Borel bireducible to a nonsmooth hyperfinite Borel equivalence relation. This answers a question of \cite{bec98}.

Figures (1)

• Figure 1: The Knaster continuum

Theorems & Definitions (2)

• Theorem 1
• proof