The class of recursively enumerable subsets of a recursively enumerable set

For any set α, let ΘA a denote the index set of the class of all recursively enumerable (r.e.) subsets of a (i.e., if {TFaJa^o is a standard enumeration of all r.e.sets, ΘA a -{x I W x c a}.) The purpose of this paper is to examine the possible Turing degrees of the sets θA a when a is r.e.It is proved that if b is any nonrecursive r.e.degree, the Turing degrees of sets ΘA a for a r.e., a e b, are exactly the degrees c > 0' such that e is r.e. in b.