文摘
We prove that for every Borel equivalence relation E, either E is Borel reducible to E0, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation E , either 2b4c1b532d2" title="Click to view the MathML source">E∈F or F is Borel reducible to E , then F consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.