arXiv subjects
Publications and source records attributed to Yusuf Uyar.
In this paper, we show that the arc-connectedness equivalence relation on a Polish subspace of the real plane is an essentially hyperfinite Borel equivalence relation. This result provides the optimal upper bound for such a Borel equivalence relation.
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.
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}.