Minimal extensions of Π 0 1 classes

A minimal extension of a Π 0 1 class P is a Π 0 1 class Q such that P ⊂ Q , Q – P is infinite, and for any Π 0 1 class R , if P ⊂ R ⊂ Q , then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q ′ have the same Cantor‐Bendixson derivative. We show that for any class P which has a single limit point A , and that point of degree ≤ 0 , P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)