{
  "id": "1412.4268",
  "version": "v1",
  "published": "2014-12-13T19:04:22.000Z",
  "updated": "2014-12-13T19:04:22.000Z",
  "title": "The strong Pytkeev property in topological spaces",
  "authors": [
    "Taras Banakh",
    "Arkady Leiderman"
  ],
  "comment": "15 pages. arXiv admin note: text overlap with arXiv:1311.1468",
  "categories": [
    "math.GN"
  ],
  "abstract": "A topological space $X$ has the strong Pytkeev property at a point $x\\in X$ if there exists a countable family $\\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\\subset X$ and subset $A\\subset X$ accumulating at $x$, there is a set $N\\in\\mathcal N$ such that $N\\subset O_x$ and $N\\cap A$ is infinite. We prove that for any $\\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\\to \\{y\\}\\subset Y$. If the space $Y$ is rectifiable, then the function space $C_k(X,Y)$ is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces $(X_n,*_n)$, $n\\in\\omega$, with the strong Pytkeev property their Tychonoff product and their small box-product both have the strong Pytkeev property at the distinguished point. We prove that a sequential rectifiable space $X$ has the strong Pytkeev property if and only if $X$ is metrizable or contains a clopen submetrizable $k_\\omega$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-13T19:04:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54E20",
      "54C35",
      "22A30"
    ],
    "keywords": [
      "strong pytkeev property",
      "topological space",
      "function space",
      "locally precompact topological group",
      "tychonoff product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.4268B"
    }
  }
}