{
  "id": "2101.02825",
  "version": "v1",
  "published": "2021-01-08T02:19:51.000Z",
  "updated": "2021-01-08T02:19:51.000Z",
  "title": "On iso-dense and scattered spaces in $\\mathbf{ZF}$",
  "authors": [
    "Kyriakos Keremedis",
    "Eleftherios Tachtsis",
    "Eliza Wajch"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2009.09526",
  "categories": [
    "math.GN"
  ],
  "abstract": "A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\\mathbf{ZF}$, in the absence of the axiom of choice, basic properties of iso-dense spaces are investigated. A new permutation model is constructed in which a discrete weakly Dedekind-finite space can have the Cantor set as a remainder. A metrization theorem for a class of quasi-metric spaces is deduced. The statement \"every compact scattered metrizable space is separable\" and several other statements about metric iso-dense spaces are shown to be equivalent to the countable axiom of choice for families of finite sets. Results concerning the problem of whether it is provable in $\\mathbf{ZF}$ that every non-discrete compact metrizable space contains an infinite compact scattered subspace are also included.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-08T02:19:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E25",
      "03E35",
      "54E35",
      "54G12",
      "54D35"
    ],
    "keywords": [
      "scattered spaces",
      "non-discrete compact metrizable space contains",
      "isolated point",
      "topological space",
      "metric iso-dense spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}