Almost complemented \(\Pi^{0}_{1}\) classes (Q2501161)

This paper explores an analogue of major subsets for \(\Pi^0_1\) classes, which leads to the definition and characterization of almost complemented \(\Pi^{0}_{1}\) classes. A \(\Pi^{0}_{1}\) class \(P\subseteq 2^{\omega}\) is called an almost complemented class if there exists a \(\Pi^{0}_{1}\) class \(Q\) such that \(P\cup Q=2^{\omega}\) and \(P\cap Q\) is finite. It is shown that if \(P\) is an almost complemented \(\Pi^{0}_{1}\) class, then \(P\) is either (1) a complemented class (no boundary points); (2) a singleton (one boundary point, finite class); (3) a bristle -- a limit of complemented classes along a single computable path, i.e. one boundary point, infinite class; (4) or a finite union of the above.