Almost computably enumerable families of sets

An almost computably enumerable family that is not Ø′- computably enumerable is constructed. Moreover, it is established that for any computably enumerable (c.e.) set A there exists afamily that is X-c.e. if and only if the set X is not A-computable. © 2008 Russian Academy of Sciences, (DoM) and London Mathematical Society, Turpion Ltd