Continuity of capping in

A set A ⊆ ω is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A , B ⊆ ω , we say that A is bounded Turing reducible to (or alternatively, weakly truth table (wtt, for short) reducible to) B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B , written A ≤ bT B . Let C bT be the structure of the c.e. bT-degrees, the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in C bT . We show that for any c.e. bT-degree b ≠ 0 , 0 ′ , there is a c.e. bT-degree a > b such that for any c.e. bT-degree x , b ∧ x = 0 if and only if a ∧ x = 0 . We prove that the analog of the Seetapun local noncappability theorem from the c.e. Turing degrees also holds in C bT . This theorem demonstrates that every b ≠ 0 , 0 ′ is noncappable with any nontrivial degree below some a > b (i.e. if x < a and x ∧ b = 0 then x = 0 ).