Descriptive complexity of countable-compactum representations

Determine the Borel or projective complexity of the set of sequences g ∈ C(2^N)^N such that B(g) is representable in C(K) for some countable compactum K.

Background

Because every B-ideal has a representation in the Polish space C(2N), the paper formulates representability in a countable compactum as a descriptive-set-theoretic classification problem. It also identifies the analogous c0-representability set as a special case.

References

Question 6.21. Find the (Borel or projective) complexity of the following set: {g ∈ C(2N)N ∶ B(g) is representable in C(K) for some countable compactum K}.

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 6.21, Section 6.3, page 40