The Complexity of Connectedness Relations on Polish Spaces

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.