{
  "abstract": "We answer four questions on irreducible determined $T_0$-topologies and a related question on monotone determined topologies posed by Xu, Yang and Luo (Topology Appl.\\ 382 (2026), 109762). For every $T_0$ topology $τ$, the family $τ_r$ used to define its irreducible-derived topology is already a topology and preserves all nonempty irreducible subsets and their closures. Consequently, the least irreducible determined refinement is obtained in one step: $τ_r=\\rd(τ)=\\Rd(τ)$. For an arbitrary poset $P$, we describe the weakest order-compatible irreducible determined topology $ρ(P)$ using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets. The Scott space of the binary-prefix domain provides a common counterexample to saturated heredity for irreducible determined spaces and monotone determined spaces: its maximal-point subspace is Cantor space. We also describe $ρ(P)$-closure by iterating an irreducible-cut operator, characterize Scott continuity of this operator, and give an example requiring two applications of the cut operator. A finite-decomposition condition extends the topology and closure-preservation results to general subset systems. For the Rudin and well-filtered determined systems, we exhibit a common $T_1$ example whose first and second derived topologies are the co-countable and discrete topologies, respectively. Thus the first derived topology need not be determined by the same subset system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04808",
  "author_total": 2,
  "authors": [
    {
      "name": "Chongyun Zhao"
    },
    {
      "name": "Xiaoquan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dbcmz5rfs9scb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04808",
  "fulltext_url": null,
  "id": "dbcmz5rfs9scb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04808",
  "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/2611.04808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dbcmz5rfs9scb.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Four problems about the irreducible determined $T_0$-topologies",
  "title_text": "Four problems about the irreducible determined T_0-topologies",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/dbcmz5rfs9scb",
  "venue": "arXiv",
  "versions": []
}