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EQUIVALENCE RELATIONS WHICH ARE BOREL SOMEWHERE
Published online by Cambridge University Press: 08 September 2017
Abstract
The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a
${\bf{\Sigma }}_1^1$ or a
${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes
${\bf{\Delta }}_1^1$. If for all
$z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$, z♯ exists, then there exists an I+
${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a
${\bf{\Delta }}_1^1$ equivalence relation.
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- Copyright © The Association for Symbolic Logic 2017
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