Covering-property characterization for (M, K(M)) ≤T K(Q)

Characterize, via a covering property analogous to the Menger property, those separable metrizable spaces M for which (M, K(M)) is Tukey-below K(Q); and ascertain whether such a characterization extends to Lindelöf spaces.

Background

For separable metrizable spaces M, Theorem 3.4 connects Menger-type covering properties to the Tukey relation (M, K(M)) ≥T (ωω, K(ωω)). The authors ask for a similar, purely topological covering characterization that precisely captures when (M, K(M)) is Tukey-below K(Q).

They also seek to understand whether such a covering characterization can be formulated and proven in the broader class of Lindelöf spaces.