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Analysis shows the arc-connectedness relation is a hyperfinite Borel equivalence in Polish subspaces, indicating its optimality.
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.
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Yusuf Uyar (2025) studied this question.
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