On $\sum^1_1$ Equivalence Relations with Borel Classes of Bounded Rank

Sami, Ramez L.

J. Symbolic Logic, Tome 49 (1984) no. 1, p. 1273-1283 / Harvested from Project Euclid

Résumé

In Baire space $\mathscr{N} = ^\omega\omega$ we define a sequence of equivalence relations $\langle E_\nu | \nu < \omega^{ck}_1 \rangle$, each $E_v$ being $\Sigma^1_1$ with classes in $\Pi^0_{1 + \nu + 1}$ and such that (i) $E_\nu$ does not have perfectly many classes, and (ii) $\mathscr{N}/E_\nu$ is countable iff $\omega^L_\nu < \omega_1$. This construction can be extended cofinally in $(\delta^1_3)^L$. A new proof is given of a theorem of Hausdorff on partitions of $\mathbf{R}$ into $\omega_1$ many $\Pi^0_3$ sets.

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Publié le : 1984-12-14
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