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

TL;DR

This paper proves that the singleton degrees within the $\Sigma^0_2$ and $\Delta^0_2$ sets are not dense by constructing specific sets, answering an open question and revealing new structural properties of these degrees.

Contribution

It demonstrates the existence of $\Delta^0_2$ sets with singleton degrees forming minimal covers, showing non-density of $\Sigma^0_2$ singleton degrees and related degrees.

Findings

$\Sigma^0_2$ singleton degrees are not dense.

$\Delta^0_2$ singleton degrees are not dense.

Constructed sets $D$ and $E$ lie in the same enumeration degree.

Abstract

Answering an open question raised by Cooper, we show that there exist $\Delta^0_2$ 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 $\Sigma^{0}_{2}$ singleton degrees, and the $\Delta^{0}_{2}$ singleton degrees, are not dense (and consequently the $\Pi^0_2$ $Q$-degrees, and the $\Delta^{0}_{2}$ $Q$-degrees, are not dense). Moreover $D$ and $E$ can be built to lie in the same enumeration degree.