{
  "abstract": "This paper solves two open problems concerning the well-filteredness of topological spaces. The first, due to Shen, Xi and Zhao\\cite{shen2023further}, asks whether a space must be open well-filtered whenever its upper space is so; the answer is negative; the Hofmann-Lawson space provides a counterexample. The second, posed by Li, Yuan and Zhao\\cite{li2020unified}, inquires whether $PF$-well-filteredness and $PF$- sobriety are equivalent under the assumption of local compactness; we answer it in the affirmative. Subsequently, we give characterizations of $PF$-sobriety and $PF$-well-filteredness for $T_0$ spaces, respectively. On the basis of the characterizations, we obtain the following result: $PF$-sobriety and $PF$-well-filteredness coincide on the Rudin spaces, which extends the previously established equivalence for locally compact spaces, as every locally compact space is a Rudin space. Furthermore, we give a counterexample to show that for nonempty $T_0$ topological spaces $X$ and $Y$, if $Y$ is a $PF$-well-filtered ($PF$-sober) space, then the function space consisting of all continuous maps from $X$ to $Y$ equipped with the Isbell topology may not be a $PF$-well-filtered ($PF$-sober) space even when $X$ is a core-compact space. We also prove that if the function space comprising all continuous maps from nonempty $X$ to $Y$ endowed with the Isbell topology is $PF$-well-filtered ($PF$-sober), then $Y$ is also $PF$-well-filtered ($PF$-sober) even though $PF$-sobriety and $PF$-well-filteredness are not preserved under retracts. Finally, we show that the reflective hull of the category of $PF$-well-filtered ($PF$-sober) spaces with continuous maps in $TOP_0$ is $TOP_0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21578",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruoxi Chen"
    },
    {
      "name": "Qingguo Li"
    },
    {
      "name": "Longchun Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5c76gfy9zyscs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21578",
  "fulltext_url": null,
  "id": "5c76gfy9zyscs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21578",
  "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": "W7222611627",
  "pdf_url": "https://arxiv.org/pdf/2610.21578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5c76gfy9zyscs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$PF$-well-filtered spaces",
  "title_text": "PF-well-filtered spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/5c76gfy9zyscs",
  "venue": "arXiv",
  "versions": []
}