Equivalence Relations Which Are Borel Somewhere

Abstract

The following will be shown: Let $I$ be a $\sigma$-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.

Made some minor changes and corrected some typos. Added some new remarks at the end. Included funding information

References in corpus (1)

• The Countable Admissible Ordinal Equivalence Relation

Cited by in corpus (1)

• Perfect Set Theorems for Equivalence Relations with $I$ - small classes