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 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 EbT 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 EbT. We show that for any c.e. bT-degree b 6= 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. This is the first continuity property of the c.e. bT-degrees.