{
  "id": "1812.10166",
  "version": "v1",
  "published": "2018-12-25T21:11:39.000Z",
  "updated": "2018-12-25T21:11:39.000Z",
  "title": "The $κ$-Fréchet--Urysohn property for locally convex spaces",
  "authors": [
    "S. Gabriyelyan"
  ],
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "A topological space $X$ is $\\kappa$-Fr\\'{e}chet--Urysohn if for every open subset $U$ of $X$ and every $x\\in \\overline{U}$ there exists a sequence in $ U$ converging to $x$. For a Tychonoff space $X$, denote by $C_p(X)$ and $C_k(X)$ the space $C(X)$ of all real-valued continuous functions on $X$ endowed with the pointwise topology and the compact-open topology, respectively. The space $X$ is Ascoli if every compact subset of $C_k(X)$ is evenly continuous. We prove that every $\\kappa$-Fr\\'{e}chet--Urysohn space $X$ is Ascoli. We apply this statement and some of the main results from [1,4-8,12] to characterize the $\\kappa$-Fr\\'echet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we show that $C_p(X)$ is Ascoli iff $X$ has the property $(\\kappa)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-25T21:11:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03",
      "46A08",
      "54C35"
    ],
    "keywords": [
      "locally convex spaces",
      "fréchet-urysohn property",
      "important classes",
      "open subset",
      "compact-open topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}