Separable quotients of pointwise function spaces and relative consistency

Abstract

Assuming Jensen's diamond principle, we construct a separable crowded compact zero-dimensional space $K$ of weight ${\aleph }_1$ and cardinality $2^{{\aleph }_1}$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. In particular, $K$ is an Efimov space: it contains neither a nontrivial convergent sequence nor a copy of $\beta \mathbb{N}$. We also prove in ZFC that the existence of a separable quotient of this kind is equivalent to the existence of a metrisable one. On the positive side, established finite-support random-sign methods give a split quotient of $C_p(K)$ onto $(c_0{)}_p$ whenever $K$ is infinite compact and $w(K)<\mathrm{cov}\mathcal{N}$, with explicit norm-one operators. Consequently, the universal quotient assertion for compacta of weight ${\aleph }_1$ is independent relative to the consistency of ZFC alone. A real-valued measurable continuum gives the positive result below the continuum and reduces the remaining problem to separable compacta of weight exactly the continuum. The counterexample uses direct diamond prediction, a simultaneous successor lemma, and a compactness argument converting localised measure limits into differences of point masses.

arXiv comment28 pp