In computability theory, a Π01 class is a subset of 2ω of a certain form.
They are also used in the application of recursion theory to other branches of mathematics (Cenzer 1999, p. 39).
An element f of 2ω is a path through a tree T on 2<ω if every finite initial segment of f is in T. A (lightface) Π01 class is a subset C of 2ω for which there is a computable tree T such that C consists of exactly the paths through T. A boldface Π01 class is a subset D of 2ω for which there is an oracle f in 2ω and a subtree tree T of 2< ω from computable from f such that D is the set of paths through T. The boldface Π01 classes are exactly the same as the closed sets of 2ω and thus the same as the boldface Π01 subsets of 2ω in the Borel hierarchy.
A subset B of 2ω is effectively closed if there is a recursively enumerable sequence ⟨σi : i ∈ ω⟩ of elements of 2< ω such that each g ∈ 2ω is in B if and only if there exists some i such that σi is an initial segment of B.
For each effectively axiomatized theory T of first-order logic, the set of all completions of T is a