{
  "abstract": "We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a $T_0$-space $X$ is stably compact---in particular, if every such subspace is Hausdorff---then $\\PS^n(X)$ is co-sober for every finite $n\\geq 1$. This applies to every $k_1$-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each $k$-irreducible compact saturated family is generated by its union.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14446",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Ji"
    },
    {
      "name": "Xiaoquan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1q1tbzbxkmnsv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14446",
  "fulltext_url": null,
  "id": "1q1tbzbxkmnsv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14446",
  "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.GN"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221787697",
  "pdf_url": "https://arxiv.org/pdf/2610.14446",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1q1tbzbxkmnsv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14446",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14446"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Compact localization and co-sobriety of Smyth power spaces",
  "title_text": "Compact localization and co-sobriety of Smyth power spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1q1tbzbxkmnsv",
  "venue": "arXiv",
  "versions": []
}