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 ℵ1 and cardinality 2ℵ1 such that Cp(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 βℕ. 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 Cp(K) onto (c0)p whenever K is infinite compact and w(K)<cov𝒩, with explicit norm-one operators. Consequently, the universal quotient assertion for compacta of weight ℵ1 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.
Related
- Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients
- Symmetric compactifications of the integers and separable quotients of spaces Cp(X)
- Three Problems on Separable Quotients of Precompact Abelian Groups
- Pointwise function spaces over compacta are not weak Banach spaces
- Hereditarily Lindelöf and hereditarily separable subspaces of function spaces
BibTeX
@misc{kania2026separable,
title = {{Separable quotients of pointwise function spaces and relative consistency}},
author = {Tomasz Kania and Jerzy Kąkol},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19351},
eprint = {2609.19351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Separable quotients of pointwise function spaces and relative consistency
AU - Tomasz Kania
AU - Jerzy Kąkol
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.19351
UR - https://arxiv.org/abs/2609.19351
AB - Assuming Jensen's diamond principle, we construct a separable crowded compact zero-dimensional space K of weight ℵ_1 and cardinality 2^ℵ_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 βℕ. 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)<cov𝒩, with explicit norm-one operators. Consequently, the universal quotient assertion for compacta of weight ℵ_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.
ER -