A nonsmooth path-connectedness relation in the real plane

TL;DR

This paper constructs a compact subset of the real plane with a path-connectedness relation that is Borel bireducible to a nonsmooth hyperfinite Borel equivalence relation, answering an open question.

Contribution

It introduces a specific compact subset of the plane with a complex path-connectedness relation, linking topology and descriptive set theory.

Findings

Path-connectedness relation is Borel bireducible to a nonsmooth hyperfinite Borel equivalence relation.

Provides a construction answering a previously open question.

Bridges concepts in topology and descriptive set theory.

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}.