{
  "id": "2404.06623",
  "version": "v1",
  "published": "2024-04-09T21:16:19.000Z",
  "updated": "2024-04-09T21:16:19.000Z",
  "title": "Quasiorders for a characterization of iso-dense spaces",
  "authors": [
    "Tom Richmond",
    "Eliza Wajch"
  ],
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in $\\mathbf{ZF}$ a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family $\\mathcal{A}$ of subsets of a set $X$, a quasiorder $\\lesssim_{\\mathcal{A}}$ on $X$ determined by $\\mathcal{A}$ is defined. Necessary and sufficient conditions for $\\mathcal{A}$ are given to have the property that the topology consisting of all $\\lesssim_{\\mathcal{A}}$-increasing sets coincides with the generalized topology on $X$ consisting of the empty set and all supersets of non-empty members of $\\mathcal{A}$. The results obtained, applied to the quasiorder $\\lesssim_{\\mathcal{D}}$ determined by the family $\\mathcal{D}$ of all dense sets of a given (generalized) topological space, lead to a new characterization of non-trivial iso-dense spaces. Independence results concerning resolvable spaces are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-09T21:16:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A05",
      "54A10",
      "54F05",
      "54F30",
      "54G12",
      "06A75",
      "06F30",
      "54A35",
      "03E35"
    ],
    "keywords": [
      "characterization",
      "independence results concerning resolvable spaces",
      "non-trivial iso-dense spaces",
      "topological space",
      "special quasiorders"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}