{
  "id": "2407.12160",
  "version": "v1",
  "published": "2024-07-16T20:32:42.000Z",
  "updated": "2024-07-16T20:32:42.000Z",
  "title": "Topological complexity of ideal limit points",
  "authors": [
    "Marek Balcerzak",
    "Szymon Glab",
    "Paolo Leonetti"
  ],
  "categories": [
    "math.GN",
    "math.CA",
    "math.FA"
  ],
  "abstract": "Given an ideal $\\mathcal{I}$ on the nonnegative integers $\\omega$ and a Polish space $X$, let $\\mathscr{L}(\\mathcal{I})$ be the family of subsets $S\\subseteq X$ such that $S$ is the set of $\\mathcal{I}$-limit points of some sequence taking values in $X$. First, we show that $\\mathscr{L}(\\mathcal{I})$ may attain arbitrarily large Borel complexity. Second, we prove that if $\\mathcal{I}$ is a $G_{\\delta\\sigma}$-ideal then all elements of $\\mathscr{L}(\\mathcal{I})$ are closed. Third, we show that if $\\mathcal{I}$ is a simply coanalytic ideal and $X$ is first countable, then every element of $\\mathscr{L}(\\mathcal{I})$ is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals $\\mathcal{I}$ for which $\\mathscr{L}(\\mathcal{I})$ contains a given set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T20:32:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ideal limit points",
      "topological complexity",
      "attain arbitrarily large borel complexity",
      "simply coanalytic ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}