The restriction of a Borel equivalence relation to a sparse set (Q1407596)

This paper differs from the usual study of Borel equivalence relations defined on a Polish space in one respect. Usually the theory of definable equivalence relations developed in descriptive set theory considers the structure of the equivalence relation on the whole space. Here, as in a paper by \textit{C. E. Uzcátegui} [Trans. Am. Math. Soc. 347, 2025-2039 (1995; Zbl 0826.03020)], the object of study is the structure of the equivalence relation on definable subsets of the Polish space. The paper focuses on the Glimm-Effros dichotomy, which states that any equivalence relation which is sparse is smooth. Let \(X\) be a Polish space and \(A \subseteq X\): \(A\) is \(E\)-sparse means that there is no one-to-one continuous function \(f: 2^{\omega} \to A\) which reduces the Vitali equivalence relation on \(2^{\omega}\) (two sequences are equivalent if they differ on finitely many coordinates) to \(E\); \(A\) is \(E\)-smooth if there exists \(C \supseteq A\) Borel and \(E\)-invariant with a countable Borel separating family for \(E \restriction C\). The dichotomy was proved for an arbitrary Borel equivalence relation on a Polish space by \textit{L. A. Harrington}, \textit{A. S. Kechris} and \textit{A. Louveau} [J. Am. Math. Soc. 3, 903-928 (1990; Zbl 0778.28011)]. The first theorem of the paper provides a counterexample (due to Kechris) to this dichotomy for \(\mathbf\Pi^1_1\) sets: there exists a countable Borel equivalence relation \(E\) and a \(\mathbf\Pi^1_1\) \(E\)-invariant set \(A\) which is \(E\)-sparse and not \(E\)-smooth. This theorem is proved in \(\mathbf{ZF + DC}\). The other results of the paper concern supersets and subsets of sparse and smooth sets. It is obvious that the sparseness of \(A\) is implied by the sparseness of its Borel (indeed, compact) subsets, while the smoothness of \(A\) is implied by the smoothness of a Borel superset of \(A\). This observation leads immediately to two questions, and in this paper both are answered within \(\mathbf{ZF + DC + AD}_{\mathbb R}\). First it is shown that the sparseness of \(A\) is not controlled by the sparseness of its ``simple'' supersets: there exists a countable Borel equivalence relation \(E\) such that for any Wadge class \(\mathbf\Lambda\) there exists an \(E\)-invariant set \(A\) which is \(E\)-sparse but has no \(E\) sparse superset in \(\mathbf\Lambda\). Then it is shown that the smoothness of an arbitrary \(A\) is implied by the smoothness of its \(\mathbf\Sigma^1_2\) subsets (it is open whether \(\mathbf\Pi^1_1\) can replace \(\mathbf\Sigma^1_2\) here, while the first theorem implies that Borel cannot).