The Complexity of Connectedness Relations on Polish Spaces

TL;DR

This paper explores the complexity of various connectedness relations on Polish spaces, revealing their interrelations and establishing the Borel reducibility of arc-connection to the Vitali relation, with implications for understanding their descriptive set-theoretic complexity.

Contribution

It systematically analyzes three connectedness relations on Polish spaces and demonstrates their interconnections and complexity classifications, including Borel reducibility results.

Findings

Arc-connection relation in the plane is Borel reducible to the Vitali equivalence relation.

Chain continuum-connection relation has higher complexity on locally compact subsets of the plane.

All studied connectedness relations are closely related and tied to the arc-connection relation.

Abstract

We systematically investigate three different equivalence relations of connectedness: being connected by arcs, being connected by continua and being connected by chains of continua of decreasing diameter. The investigation is conducted from the point of view of Borel reductions, mainly on Polish spaces. All of the studied equivalence relations turn out to be tied together and intimately related to the arc-connection relation. Among other results, it is shown that the arc-connection relation in the plane is Borel reducible to the Vitali equivalence relation and thus of a very low complexity. The same is proven for the chain continuum-connection relation on locally compact subsets of the plane, on which the continuum-connection relation is shown to have higher complexity.