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

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