{
  "id": "2407.18618",
  "version": "v1",
  "published": "2024-07-26T09:22:39.000Z",
  "updated": "2024-07-26T09:22:39.000Z",
  "title": "Characterizing function spaces which have the property (B) of Banakh",
  "authors": [
    "Mikołaj Krupski",
    "Kacper Kucharski",
    "Witold Marciszewski"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "A topological space $Y$ has the property (B) of Banakh if there is a countable family $\\{A_n:n\\in \\mathbb{N}\\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space $X$ has the following property $(\\kappa)$: every sequence of disjoint finite subsets of $X$ has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces $X$ whose all bounded subsets are finite, yet $X$ fails to have the property $(\\kappa)$. This answers a question of Tkachuk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-26T09:22:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "54E52",
      "54A10"
    ],
    "keywords": [
      "characterizing function spaces",
      "tychonoff space",
      "disjoint finite subsets",
      "point-finite open expansion",
      "dense subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}