Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π 0 1 classes, Annals of Pure and Applied Logic 59 (1993) 79–139. A Π 0 1 class P ⊂ {0, 1} ω is thin if every Π 0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π 0 1 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π 0 1 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no set of degree ≥ 0” can be a member of any thin Π 0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π 0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π 0 1 class. It is shown that no maximal set can have rank one in any Π 0 1 class, while there exist maximal sets of rank 2. The connection between Π 0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π 0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥ 0”.