{
  "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 $β\\mathbb N$. 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 $w(K)<\\operatorname{cov}\\mathcal N$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19351",
  "author_total": 2,
  "authors": [
    {
      "name": "Tomasz Kania"
    },
    {
      "name": "Jerzy Kąkol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/175cdd8sp88nn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19351",
  "fulltext_url": null,
  "id": "175cdd8sp88nn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19351",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213581278",
  "pdf_url": "https://arxiv.org/pdf/2609.19351",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/175cdd8sp88nn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.19351",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19351"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Separable quotients of pointwise function spaces and relative consistency",
  "title_text": "Separable quotients of pointwise function spaces and relative consistency",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/175cdd8sp88nn",
  "venue": "arXiv",
  "versions": []
}