Some results on projective equivalence relations

• Main Author: Li, Xuhua
• Format: Others
• Language: en
• Published: 1998
• Online Access: https://thesis.library.caltech.edu/10016/1/Li_X_1998.pdf; Li, Xuhua (1998) Some results on projective equivalence relations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/pn2n-7z61. https://resolver.caltech.edu/CaltechTHESIS:01202017-113604553 <https://resolver.caltech.edu/CaltechTHESIS:01202017-113604553>

<p>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].</p> <p>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,</p> <p>xEy ↔ (p(x),p(y))∈ e </p> <p>This lifts Harrington's result about thin ∏¹₁ equivalence relations to thin ∏¹₃ equivalence relations.</p>