Efimov spaces and separable quotients of C_p(K)

Determine whether any of the consistently constructed Efimov compact spaces—compact spaces containing neither a convergent sequence nor a copy of the Čech–Stone compactification βω—has a function space C_p(K) without an infinite-dimensional separable quotient.

Background

The paper reduces the general Separable Quotient Problem for spaces C_p(K), with K compact, to totally asymmetric compactifications of the integers. It also observes that a compact space K for which C_p(K) lacks a separable quotient must be an Efimov space.

Efimov spaces of the relevant type have so far been constructed only consistently. The unresolved issue is whether any such known example actually realizes the failure of separable quotients for C_p(K); the paper's reduction does not settle that existence question.

References

It is not known whether any of those examples $K$ is such that $C_p(K)$ does not admit a separable quotient, see e.g. for details.

— Symmetric compactifications of the integers and separable quotients of spaces $C_p(X)$  (2609.18904 - Silber et al., 16 Sep 2026) in Section 1, Introduction (paragraph following Theorem mainD)

That construction leaves the existence of an arbitrary infinite-dimensional metrisable quotient of $C_p(T)$ open.

— A consistent failure of separable quotients for pointwise function spaces  (2609.19351 - Kania et al., 16 Sep 2026) in Section 1, Introduction