Some results on projective equivalence relations

Author: Li, Xuhua

Year: 1998

Degree: Dissertation (Ph.D.)

Advisor: Kechris, Alexander S.

Committee Members: Ramakrishnan, Dinakar; Luxemburg, W. A. J.

Option: Mathematics

Abstract

We construct a ∏¹₁ equivalence relation E on ω^ω for which there is no largest E-thin, E-invariant ∏¹₁ subset of ω^ω. Then we lift our result to the general case. Namely, we show that there is a ∏¹_2n+1 equivalence relation for which there is no largest E-thin, E-invariant ∏¹_2n+1 set under projective determinacy. This answers an open problem raised in Kechris [Ke2].

Our second result in this thesis is a representation for thin ∏¹₃ equivalence relations on u_ω. Precisely, we show that for each thin ∏¹₃ equivalence relation E on u_ω, there is a Δ¹₃ in the codes map p: ω^ω → u_ω and a ∏¹₃ in the codes equivalence relation e on u_ω such that for all real numbers x and y,

xEy ↔ (p(x),p(y))∈ e

This lifts Harrington's result about thin ∏¹₁ equivalence relations to thin ∏¹₃ equivalence relations.

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