Pairs strictly Tukey-below K(Q)

Determine whether there exist separable metrizable spaces M such that the pair (M, K(M)) is strictly Tukey-below K(Q), i.e., (M, K(M)) <T K(Q).

Background

The authors analyze the position of K(Q) in the Tukey order and provide constraints and characterizations: if (M, K(M)) ≥T K(Q) then M is not hereditarily Baire (Proposition 3.9), and K(Q) ≥T (M, K(M)) if and only if M is 2^2 (Proposition 3.10). They also construct examples Tukey-incomparable with K(Q) (Example 3.11).

Despite these partial results, it remains unsettled whether there exist nontrivial pairs (M, K(M)) that lie strictly below K(Q), motivating further exploration of the structure just beneath K(Q) in the Tukey hierarchy.