{
  "id": "1607.03599",
  "version": "v1",
  "published": "2016-07-13T06:10:10.000Z",
  "updated": "2016-07-13T06:10:10.000Z",
  "title": "Topological spaces with a local $ω^ω$-base have the strong Pytkeev$^*$ property",
  "authors": [
    "Taras Banakh"
  ],
  "comment": "10 pages. arXiv admin note: text overlap with arXiv:1412.4268",
  "categories": [
    "math.GN"
  ],
  "abstract": "Modifying the known definition of a Pytkeev network, we introduce a notion of Pytkeev$^*$ network and prove that a topological space has a countable Pytkeev network at a point $x\\in X$ if and only if $X$ is countably tight at $x$ and has a countable Pykeev$^*$ network at $x$. We define a topological space $X$ to have the strong Pytkeev$^*$ property if $X$ has a Pytkeev$^*$ network at each point. Our main theorem says that a topological space $X$ has a countable Pytkeev$^*$ network at point $x\\in X$ if $X$ has a local $\\omega^\\omega$-base at $x$ (i.e., a neighborhood base $(U_\\alpha)_{\\alpha\\in\\omega^\\omega}$ at $x$ such that $U_\\beta\\subset U_\\alpha$ for all $\\alpha\\le\\beta$ in $\\omega^\\omega$). Consequently, each countably tight space with a local $\\omega^\\omega$-base has the strong Pytkeev property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-13T06:10:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D70",
      "54E18"
    ],
    "keywords": [
      "topological space",
      "strong pytkeev property",
      "main theorem says",
      "countable pytkeev network",
      "countably tight space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}