The singleton degrees of the ${\Sigma }_2^0$ sets are not dense

Abstract

Answering an open question raised by Cooper, we show that there exist ${\Delta }_2^0$ sets $D$ and $E$ such that the singleton degree of $E$ is a minimal cover of the singleton degree of $D$. This shows that the ${\Sigma }_2^0$ singleton degrees, and the ${\Delta }_2^0$ singleton degrees, are not dense (and consequently the ${\Pi }_2^0$ $Q$-degrees, and the ${\Delta }_2^0$ $Q$-degrees, are not dense). Moreover $D$ and $E$ can be built to lie in the same enumeration degree.