{
  "abstract": "A $T_0$-space is called $ω$-sober if every irreducible closed subset having a countable dense subset is the closure of a unique point. This is precisely $H$-sobriety for the countable irreducible subset system $\\mathcal R^ω$. We give two natural Hofmann--Mislove representations in the general $H$-sobriety framework: filters induced by $H$-sets of $X$ characterize $H$-sobriety, whereas filters induced by $H$-sets of the Smyth power space characterize super $H$-sobriety. Their relation is expressed by property $Q$. A minimal closed set argument with a countable selection proves that $\\mathcal R^ω$ has property $Q$. Consequently, $X$ is $ω$-sober if and only if its Smyth power space is $ω$-sober, equivalently, every countable irreducible compact-family filter has a compact saturated representation. We also obtain equivalent recovery, compact-intersection and representation conditions for an enlarged class containing the proper Scott-open filters with countable open bases and the filters induced by countable irreducible subsets. A further equivalent formulation uses traces on countable subspaces. The filtered-family specializations recover well-filteredness and $ω$-well-filteredness. Explicit examples show that $ω$-sobriety strictly implies $ω$-well-filteredness, which strictly implies representation for countable open bases alone.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05214",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoquan Xu"
    },
    {
      "name": "Xinjie Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d2899a3j61fbx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05214",
  "fulltext_url": null,
  "id": "d2899a3j61fbx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05214",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05214",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d2899a3j61fbx.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05214",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05214"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "H-sober spaces and Hofmann-Mislove type theorems",
  "title_text": "H-sober spaces and Hofmann-Mislove type theorems",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/d2899a3j61fbx",
  "venue": "arXiv",
  "versions": []
}