{
  "id": "1806.10177",
  "version": "v1",
  "published": "2018-06-26T19:18:10.000Z",
  "updated": "2018-06-26T19:18:10.000Z",
  "title": "Compact complement topologies and k-spaces",
  "authors": [
    "Kyriakos Keremedis",
    "Cenap Öcel",
    "Artur Piękosz",
    "Mohammed Al Shumrani",
    "Eliza Wajch"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $(X,\\tau)$ be a Hausdorff space, where $X$ is an infinite set. The compact complement topology $\\tau^{\\star}$ on $X$ is defined by: $\\tau^{\\star}=\\{\\emptyset\\} \\cup \\{X\\setminus M, \\text{where $M$ is compact in $(X,\\tau)$}\\}$. In this paper, properties of the space $(X, \\tau^{\\star})$ are studied in $\\mathbf{ZF}$ and applied to a characterization of $k$-spaces, to the Sorgenfrey line, to some statements independent of $\\mathbf{ZF}$, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Among other results, it is proved that the axiom of countable multiple choice (\\textbf{CMC}) is equivalent with each of the following two sentences: (i) every Hausdorff first countable space is a $k$-space, (ii) every metrizable space is a $k$-space. A \\textbf{ZF}-example of a countable metrizable space whose compact complement topology is not first countable is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-26T19:18:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D50",
      "54D55",
      "54A35",
      "54E99",
      "54D30",
      "54E35"
    ],
    "keywords": [
      "compact complement topology",
      "metrizable space",
      "hausdorff first countable space",
      "hausdorff space",
      "partial topologies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}