{
  "abstract": "Every neighborhood assignment on a charming space $X$ has a countable kernel whose closure in $X$ is Lindelöf $Σ$. Consequently, for any two neighborhood assignments $M$ and $N$ on $X$, there is a closed discrete set $D\\subseteq X$ such that $M[N[D]]=X$. We give an exact one-step formulation of the remaining $D$-space problem and identify the universal pointwise square-refinement condition with the classical $z$-space condition. For continuous closed surjections, we prove a fixed-assignment lifting equivalence. We also show that every uncountable $\\mathrm{HFC}_{\\mathrm w}$ subspace of $[ω_1]^{\\leqω}$ is dually closed countable but not charming. For a fixed Lindelöf $Σ$ kernel $Y$, we iterate the Hu--Yin boundary reduction by a transfinite closed-core process. At every stage, the current iterate is a $D$-space if and only if $X$ is a $D$-space. The iteration stabilizes before $hL(Y)^+$ at the largest closed subspace in which the kernel trace and its complement are both dense. The bound is sharp: for every infinite cardinal $κ$ and every $γ<κ^+$, there is a compact example with a dense $σ$-compact Lindelöf $Σ$ kernel $Y$ satisfying $hL(Y)=|Y|=κ$ whose first stabilization stage is $γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35909",
  "author_total": 1,
  "authors": [
    {
      "name": "Xing-Yu Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3mk16q6vsjs3n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35909",
  "fulltext_url": null,
  "id": "3mk16q6vsjs3n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35909",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.35909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3mk16q6vsjs3n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35909"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Closed Lindelöf $Σ$ Kernels of Neighborhood Assignments in Charming Spaces",
  "title_text": "Closed Lindelöf Σ Kernels of Neighborhood Assignments in Charming Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/3mk16q6vsjs3n",
  "venue": "arXiv",
  "versions": []
}