{
  "id": "1503.08359",
  "version": "v1",
  "published": "2015-03-28T22:15:26.000Z",
  "updated": "2015-03-28T22:15:26.000Z",
  "title": "On The Number Of Topologies On A Finite Set",
  "authors": [
    "Muhammet Yasir Kızmaz"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We denote the number of distinct topologies which can be defined on the set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime $p$, $T(p^k)\\equiv k+1 \\ mod \\ p$, and that for each non-negative integer $n$ there exists a unique $k$ such that $T(p+n)\\equiv k$. We calculate $k$ for $n=1,2,3,4$. We give an elementary proof for a result of Z.I.Borevich to the effect that $T_0(p+n)\\equiv T_0(n+1) \\ mod \\ p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-28T22:15:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50"
    ],
    "keywords": [
      "finite set",
      "distinct topologies",
      "elementary proof",
      "non-negative integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}