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arXiv · 1605.06051

Measurability and Perfect Set Theorems for Equivalence Relations with Small Classes

Ohad Drucker
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Abstract

We ask whether $\mathbf{Δ^1_2}$ or $\mathbf{Σ^1_2}$ equivalence relations with $I$-small classes for $I$ a $σ$-ideal must have perfectly many classes. We show that for a wide class of ccc $σ$-ideals, a positive answer for $\mathbf{Δ^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{Δ^1_2}$ sets. However, the analogous statement for $\mathbf{Σ^1_2}$ equivalence relations is false: $\mathbf{Σ^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if $\mathbf{Δ^1_2}$ sets have the Baire property.

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Ohad Drucker. 2016-05-30. Measurability and Perfect Set Theorems for Equivalence Relations with Small Classes. https://arxiv.org/abs/1605.06051

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