{
  "id": "1810.12998",
  "version": "v1",
  "published": "2018-10-30T20:44:46.000Z",
  "updated": "2018-10-30T20:44:46.000Z",
  "title": "Cardinal inequalities for $S(n)$-spaces",
  "authors": [
    "Ivan S. Gotchev"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Hajnal and Juh\\'asz proved that if $X$ is a $T_1$-space, then $|X|\\le 2^{s(X)\\psi(X)}$, and if $X$ is a Hausdorff space, then $|X|\\le 2^{c(X)\\chi(X)}$ and $|X|\\le 2^{2^{s(X)}}$. Schr\\\"oder sharpened the first two estimations by showing that if $X$ is a Hausdorff space, then $|X|\\le 2^{Us(X)\\psi_c(X)}$, and if $X$ is a Urysohn space, then $|X|\\le 2^{Uc(X)\\chi(X)}$. In this paper, for any positive integer $n$ and some topological spaces $X$, we define the cardinal functions $\\chi_n(X)$, $\\psi_n(X)$, $s_n(X)$, and $c_n(X)$, called respectively $S(n)$-character, $S(n)$-pseudocharacter, $S(n)$-spread, and $S(n)$-cellularity, and using these new cardinal functions we show that the above-mentioned inequalities could be extended to the class of $S(n)$-spaces. We recall that the $S(1)$-spaces are exactly the Hausdorff spaces and the $S(2)$-spaces are exactly the Urysohn spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-30T20:44:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54D10"
    ],
    "keywords": [
      "cardinal inequalities",
      "hausdorff space",
      "urysohn space",
      "cardinal functions",
      "topological spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}