[Paper Review] Equivalence Relations Which Are Borel Somewhere

William Chan|arXiv (Cornell University)|Nov 25, 2015

Advanced Topology and Set Theory9 references3 citations

TL;DR

This paper investigates when analytic or coanalytic equivalence relations on Polish spaces can be restricted to a $Δ_1^1$ set where the restriction becomes a $Δ_1^1$ equivalence relation. Using forcing and absoluteness in the context of $̼$-ideals for which the associated forcing is proper, the authors show that under the assumption that $z^\sharp$ exists for all $z$ of size $(2^{\aleph_0})^+$, such a $Δ_1^1$ set $C$ exists. The key contribution is a positive answer to a refined version of the main question for $Σ_1^1$ and $Π_1^1$ equivalence relations with all classes $Δ_1^1$, under large cardinal assumptions.

ABSTRACT

The following will be shown: Let $I$ be a $σ$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence relation on this Polish space with all equivalence classes Borel. If sharps of certain sets exist, then there is an $I^+$ Borel subset $C$ of $X$ such that $E \upharpoonright C$ is a Borel equivalence relation.

Motivation & Objective

• To determine under what conditions an analytic or coanalytic equivalence relation on a Polish space can be restricted to a $Δ_1^1$ set where the restriction becomes a $Δ_1^1$ equivalence relation.
• To address the robustness of the main question by requiring all equivalence classes to be $Δ_1^1$, avoiding trivialities from countable sets.
• To use the forcing-theoretic framework of $σ$-ideals with proper associated forcing $Π_I$ to derive absoluteness and genericity arguments.
• To establish the existence of a $Δ_1^1$ set $C$ such that $E\upharpoonright C$ is $Δ_1^1$, under the assumption that $z^\sharp$ exists for all $z$ of size $(2^{\aleph_0})^+$.
• To explore the existence of $Π_1^1$ equivalence relations whose classes are uncountable thin sets in $L$, particularly in the context of constructible reals and forcing extensions.

Proposed method

• Utilizes the forcing $Π_I$ consisting of $I^+$ $Δ_1^1$ subsets of a Polish space $X$, ordered by inclusion, where $I$ is a $σ$-ideal.
• Applies Zapletal's characterization that $Π_I$ is proper if and only if for every countable $M \prec H_\Theta$ with $\u03a0_I, B \in M$, the set of $Π_I$-generic elements over $M$ is $I^+$ and $Δ_1^1$.
• Employs absoluteness and genericity to transfer properties of $E$ from the ground model to generic extensions, ensuring that $E\upharpoonright C$ remains $Δ_1^1$.
• Uses the Lusin-Novikov uniformization theorem to construct $Δ_1^1$ functions that code representations of elements in free abelian groups generated by $Δ_1^1$ sets.
• Constructs a $Π_1^1$ coset equivalence relation on $\mathbb{R}$ using a thin $Δ_1^1$ subgroup of reals, leveraging the constructibility of such sets in $L$.
• Applies the Mansfield-Solovay theorem to ensure that thin $Δ_1^1$ sets of reals remain unchanged in forcing extensions, preserving their uncountable thinness in $L$.

Experimental results

Research questions

• RQ1 Given a $σ$-ideal $I$ on a Polish space $X$ such that the forcing $\u03a0_I$ is proper, and an $\u03a3_1^1$ equivalence relation $E$ with all classes $\u0394_1^1$, does there exist a $I^+$ $\u0394_1^1$ set $C$ such that $E\upharpoonright C$ is $\u0394_1^1$?
• RQ2 For a $\u03c3$-ideal $I$ with proper $\u03a0_I$, and a $\u03a0_1^1$ equivalence relation $E$ with all classes $\u0394_1^1$, does there exist a $I^+$ $\u0394_1^1$ set $C$ such that $E\upharpoonright C$ is $\u0394_1^1$?
• RQ3 In $L$, does there exist a $\u03a0_1^1$ equivalence relation $E$ such that every equivalence class $[x]_E$ is uncountable thin?
• RQ4 Can a $\u03a0_1^1$ equivalence relation be constructed without explicit reference to a thin $\u0394_1^1$ set, so that it witnesses a positive answer to the existence of such $C$ in models with $\omega_1^L < \omega_1^V$?
• RQ5 Is it possible to partition $ {}^\omega\omega $ into $\u03a0_1^1$ pieces, each uncountable thin, via a $\u03a0_1^1$ equivalence relation in $L$?

Key findings

• Under the assumption that $z^\sharp$ exists for all $z \in H_{(2^{\aleph_0})^+}$, there exists a $\u0394_1^1$ set $C \subseteq X$ such that $E\upharpoonright C$ is $\u0394_1^1$, for any $\u03a3_1^1$ or $\u03a0_1^1$ equivalence relation $E$ on a Polish space $X$ with all classes $\u0394_1^1$.
• The existence of such a $C$ is guaranteed when the associated forcing $\u03a0_I$ is proper, using genericity and absoluteness to preserve the $\u0394_1^1$ nature of $E\upharpoonright C$.
• A $\u03a0_1^1$ equivalence relation $E$ on $\mathbb{R}$ is constructed such that each equivalence class $[x]_E$ is in bijection with a $\u0394_1^1$ uncountable thin set in $L$, specifically the subgroup generated by a $\u0394_1^1$ thin set of reals.
• The constructed equivalence relation $E$ is $\u03a0_1^1$ and satisfies $L \models \text{for all } x, [x]_E \text{ is uncountable thin}$, providing a positive answer to Question 10.8.
• The equivalence relation $E$ from Theorem 11.1 does not satisfy the requirements of Question 10.7, as it relies on a constructibly coded thin set, and fails to preserve uncountable thinness in models with $\omega_1^L < \omega_1^V$.
• It is shown that any solution to Question 10.7 must avoid explicit use of a thin $\u0394_1^1$ set, as such constructions fail to maintain the desired properties in forcing extensions with larger $\omega_1$.