{
  "abstract": "For a $T_1$-space $X$, let $Cld(X)$ denote all its nonempty closed subsets and $T_4(X)=\\{(F_1,F_2)\\in Cld(X)^2:F_1\\cap F_2=\\emptyset\\}$. In terms of closed sets, $X$ is $T_4$ iff for each $(F_1,F_2)\\in T_4(X)$, there is a pair of closed sets $ϕ_1(F_1,F_2)$ and $ϕ_2(F_1,F_2)$ such that their union is $X$ and $F_j\\cap ϕ_j(F_1,F_2)=\\emptyset$ for $j=1,2$. Thus, in this paper, for a topology $τ$ on $Cld(X)$, we introduce the definition: A $T_1$-space $X$ is called $CT_4$ with respect to $τ$ if the above maps $ϕ_j:T_4(X)\\to Cld(X)$ are continuous on $τ$. Similarly, for $i=1,2,3$, we can define a $T_1$-space to be $CT_i$ for $τ$. We only consider the Vietoris topology on $Cld(X)$ and show that every $CT_4$-space is countably compact, and every $CT_3$-space is a Fréchet-Urysohn space, every separable subspace of a $CT_3$-space is metrizable. We give relevant examples. Any finite-dimensional cubes, the infinite-dimensional cube, any finite-dimensional spheres, and all 0-dimensional compact metrizable spaces are $CT_4$. All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are $CT_3$ but not $CT_4$. Also, each metrizable space with a unique non-isolated point is $CT_3$, and is $CT_4$ if it is compact. Moreover, the infinite sum of $CT_3$ spaces is $CT_3$ but not $CT_4$. Every countable space with a unique non-isolated point is $CT_2$, and it is $CT_3$ if and only if it is metrizable. All subfields of real numbers and their complement spaces are $CT_2$ but not $CT_4$; and their $CT_3$ status remains unclear. All uncountable ordinal spaces are not $CT_2$. The one-point compactification of any uncountable discrete space is not $CT_2$. $CT_1$ and $T_1$ are equivalent, hence all spaces above are $CT_1$. Open Problems: is there a non-metrizable $CT_3$ or $CT_4$ space? Is any compact $CT_2$ space $CT_3$ or $CT_4$?",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13086",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhongqiang Yang"
    },
    {
      "name": "Nada Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/616re3se4pb3q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13086",
  "fulltext_url": null,
  "id": "616re3se4pb3q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13086",
  "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": "W7203491593",
  "pdf_url": "https://arxiv.org/pdf/2608.13086",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/616re3se4pb3q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.13086",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13086"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Axioms of Continuous Separation",
  "title_text": "Axioms of Continuous Separation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/616re3se4pb3q",
  "venue": "arXiv",
  "versions": []
}