{
  "id": "2309.14632",
  "version": "v1",
  "published": "2023-09-26T03:14:27.000Z",
  "updated": "2023-09-26T03:14:27.000Z",
  "title": "A bound for the density of any Hausdorff space",
  "authors": [
    "Nathan Carlson"
  ],
  "comment": "6 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "We show, in a certain specific sense, that both the density and the cardinality of a Hausdorff space are related to the \"degree\" to which the space is nonregular. It was shown by Sapirovskii that $d(X)\\leq\\pi\\chi(X)^{c(X)}$ for a regular space $X$ and the author observed this holds if the space is only quasiregular. We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree $nq(X)$, which is countable when $X$ is quasiregular, and showing $d(X)\\leq\\pi\\chi(X)^{c(X)nq(X)}$ for any Hausdorff space $X$. This demonstrates that the degree to which a space is nonquasiregular has a fundamental and direct connection to its density and, ultimately, its cardinality. Importantly, if $X$ is Hausdorff then $nq(X)$ is \"small\" in the sense that $nq(X)\\leq\\psi_c(X)$. This results in a unified proof of both Sapirovskii's density bound for regular spaces and Sun's bound $\\pi\\chi(X)^{c(X)\\psi_c(X)}$ for the cardinality of a Hausdorff space $X$. A consequence is an improved bound for the cardinality of a Hausdorff space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-26T03:14:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54D10"
    ],
    "keywords": [
      "hausdorff space",
      "regular space",
      "cardinality",
      "sapirovskiis density bound",
      "nonquasiregularity degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}