{
  "id": "2301.06220",
  "version": "v1",
  "published": "2023-01-16T00:24:19.000Z",
  "updated": "2023-01-16T00:24:19.000Z",
  "title": "New bounds on the cardinality of Hausdorff spaces and regular spaces",
  "authors": [
    "Nathan Carlson"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Using weaker versions of the cardinal function $\\psi_c(X)$, we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve $\\psi_c(X)$ nor its variants at all. For example, we show if $X$ is regular then $|X|\\leq 2^{c(X)^{\\pi\\chi(X)}}$ and $|X|\\leq 2^{c(X)\\pi\\chi(X)^{ot(X)}}$, where the cardinal function $ot(X)$, introduced by Tkachenko, has the property $ot(X)\\leq\\min\\{t(X),c(X)\\}$. It follows from the latter that a regular space with cellularity at most $\\mathfrak{c}$ and countable $\\pi$-character has cardinality at most $2^\\mathfrak{c}$. For a Hausdorff space $X$ we show $|X|\\leq 2^{d(X)^{\\pi\\chi(X)}}$, $|X|\\leq d(X)^{\\pi\\chi(X)^{ot(X)}}$, and $|X|\\leq 2^{\\pi w(X)^{dot(X)}}$, where $dot(X)\\leq\\min\\{ot(X),\\pi\\chi(X)\\}$. None of these bounds involve $\\psi_c(X)$ or $\\psi(X)$. By introducing the cardinal functions $w\\psi_c(X)$ and $d\\psi_c(X)$ with the property $w\\psi_c(X)d\\psi_c(X)\\leq\\psi_c(X)$ for a Hausdorff space $X$, we show $|X|\\leq\\pi\\chi(X)^{c(X)w\\psi_c(X)}$ if $X$ is regular and $|X|\\leq\\pi\\chi(X)^{c(X)d\\psi_c(X)w\\psi_c(X)}$ if $X$ is Hausdorff. This improves results of Sapirovskii and Sun. It is also shown that if $X$ is Hausdorff then $|X|\\leq 2^{d(X)w\\psi_c(X)}$, which appears to be new even in the case where $w\\psi_c(X)$ is replaced with $\\psi_c(X)$. Compact examples show that $\\psi(X)$ cannot be replaced with $d\\psi_c(X)w\\psi_c(X)$ in the bound $2^{\\psi(X)}$ for the cardinality of a compact Hausdorff space $X$. Likewise, $\\psi(X)$ cannot be replaced with $d\\psi_c(X)w\\psi_c(X)$ in the Arhangel'skii-Sapirovskii bound $2^{L(X)t(X)\\psi(X)}$ for the cardinality of a Hausdorff space $X$. Finally, we make several observations concerning homogeneous spaces in this connection.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-16T00:24:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54D10",
      "54D30",
      "54D45"
    ],
    "keywords": [
      "regular space",
      "cardinality",
      "cardinal function",
      "compact hausdorff space",
      "weaker versions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}