Abstract We study the structure of the c.e. $Q$- and $sQ$-degrees. For both structures, we show that there are join irreducible degrees but no Ahmad pairs. We show that the structures are not distributive, and that the lattice $N_{5}$ embeds in both structures. On the other hand the lattice $M_{5}$ cannot be embedded in the c.e. $sQ$-degrees, but a critical triple can be embedded. Finally, we show that no initial segment of the c.e. $sQ$-degrees or the c.e. $Q$-degrees is a lattice.

A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.