The Non-isolating Degrees Are Upwards Dense in the Computably Enumerable Degrees

The existence of isolated degrees was proved by Cooper and Yi in 1995 in [6], where a d.c.e. degree d is isolated by a c.e. degree a if a < d is the greatest c.e. degree below d. A computably enumerable degree c is non-isolating if no d.c.e. degree above c is isolated by c. Obviously, 0 is a non-isolating degree. Cooper and Yi asked in [6] whether there is a nonzero non-isolating degree. Arslanov et al. showed in [3] that nonzero non-isolating degrees exist and that these degrees are downwards dense in the c.e. degrees and can also occur in every jump class. In [10], Salts proved that there is an interval of computably enumerable degrees, each of which isolates a d.c.e. degree. Recently, Cenzer et al. [4] proved that such intervals are dense in the computably enumerable degrees, and hence the non-isolating degrees are nowhere dense in the computably enumerable degrees. In this paper, using a different type of construction to that of [3], we prove that the non-isolating degrees are upwards dense in the computably enumerable degrees. In the context of [4], this is the best possible such result.