The nonisolating degrees are nowhere dense in the computably enumerable degrees

The d.c.e. degrees were first studied by Cooper [5] and Lachlan who showed that there is a proper d.c.e degree, a d.c.e. degree containing no c.e. sets, and that every nonzero d.c.e. degree bounds a nonzero c.e. degree, respectively. The main motivation of research on the d.c.e. degrees is to study the differences between the structures of d.c.e. degrees and Δ2 degrees, and between the structures of d.c.e. degrees and c.e. degrees. Lachlan’s observation mentioned above shows that downwards density holds in the d.c.e. degrees. Hence, as Jockusch noticed, the d.c.e. degrees are not comple∗Cenzer is partially supported by National Science Foundation grants DMS 0532644 and 0554841. †Wu is partially supported by an AcRF research grant No. RG58/06 from MOE of Singapore.