{
  "id": "2211.09994",
  "version": "v1",
  "published": "2022-11-18T02:46:38.000Z",
  "updated": "2022-11-18T02:46:38.000Z",
  "title": "On $k$-ranks of topological spaces",
  "authors": [
    "Mengjie Jin",
    "Qingguo Li"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper, the concepts of $K$-subset systems and $k$-well-filtered spaces are introduced, which provide another uniform approach to $d$-spaces, $s$-well-filtered spaces (i.e., $\\mathcal{U}_{S}$-admissibility) and well-filtered spaces. We prove that the $k$-well-filtered reflection of any $T_{0}$ space exists. Meanwhile, we propose the definition of $k$-rank, which is an ordinal that measures how many steps from a $T_{0}$ space to a $k$-well-filtered space. Moreover, we derive that for any ordinal $\\alpha$, there exists a $T_{0}$ space whose $k$-rank equals to $\\alpha$. One immediate corollary is that for any ordinal $\\alpha$, there exists a $T_{0}$ space whose $d$-rank (respectively, $wf$-rank) equals to $\\alpha$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-18T02:46:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H99"
    ],
    "keywords": [
      "topological spaces",
      "well-filtered space",
      "subset systems",
      "uniform approach",
      "rank equals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}