The classification of smooth equivalence relations is accomplished by two exercises from Su Gao’s book, Invariant Descriptive Set Theory.

**Exercise 5.4.1, Su Gao, p 132.**

Show that for any equivalence relation on a Polish space , iff there are infinitely many -equivalence classes.

Suppose . So there exists Borel such that iff . Since for each , the element lies in a different class, there are infinitely many classes.

Suppose there are infinitely many -classes; enumerate a countable subset: . By the Axiom of Choice, for each pick a , and define by . Since has the discrete topology, is cts, and so $latex \text{id}(\omega) \le_B E$.

**Exercise 5.4.2, Su Gao, p 132**

Show that if an equivalence relation is smooth then either , or , or for some finite , .

By Theorem 5.4.2, if is smooth, either (1) or (2) .

So assume (2). Then either has infinitely many classes or it does not.

If it does, by Exercise 5.4.1, , hence .

If it does not, suppose has classes. List the classes: . Since is smooth, each is Borel.

Then define by iff . Then is Borel. Also , since has discrete topology and we can pick

a point in each equivalence class to map to .