Continuity of capping in CbT

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≤bTB. Let CbT 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 CbT. 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 CbT. This theorem demonstrates that every b≠0,0′ is noncappable with any nontrivial degree below some a>b (i.e. if x更多