{
  "id": "2411.10866",
  "version": "v1",
  "published": "2024-11-16T19:22:27.000Z",
  "updated": "2024-11-16T19:22:27.000Z",
  "title": "Borel complexity of sets of ideal limit points",
  "authors": [
    "Rafal Filipow",
    "Adam Kwela",
    "Paolo Leonetti"
  ],
  "categories": [
    "math.GN",
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $X$ be an uncountable Polish space and let $\\mathcal{I}$ be an ideal on $\\omega$. A point $\\eta \\in X$ is an $\\mathcal{I}$-limit point of a sequence $(x_n)$ taking values in $X$ if there exists a subsequence $(x_{k_n})$ convergent to $\\eta$ such that the set of indexes $\\{k_n: n \\in \\omega\\}\\notin \\mathcal{I}$. Denote by $\\mathscr{L}(\\mathcal{I})$ the family of subsets $S\\subseteq X$ such that $S$ is the set of $\\mathcal{I}$-limit points of some sequence taking values in $X$ or $S$ is empty. In this paper, we study the relationships between the topological complexity of ideals $\\mathcal{I}$, their combinatorial properties, and the families of sets $\\mathscr{L}(\\mathcal{I})$ which can be attained. On the positive side, we provide several purely combinatorial (not dependind on the space $X$) characterizations of ideals $\\mathcal{I}$ for the inclusions and the equalities between $\\mathscr{L}(\\mathcal{I})$ and the Borel classes $\\Pi^0_1$, $\\Sigma^0_2$, and $\\Pi^0_3$. As a consequence, we prove that if $\\mathcal{I}$ is a $\\Pi^0_4$ ideal then exactly one of the following cases holds: $\\mathscr{L}(\\mathcal{I})=\\Pi^0_1$ or $\\mathscr{L}(\\mathcal{I})=\\Sigma^0_2$ or $\\mathscr{L}(\\mathcal{I})=\\Sigma^1_1$ (however we do not have an example of a $\\Pi^0_4$ ideal with $\\mathscr{L}(\\mathcal{I})=\\Sigma^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\\mathcal{I}$ for which $\\mathscr{L}(\\mathcal{I})=\\Sigma^1_1$. On the negative side, we show that there are no ideals $\\mathcal{I}$ such that $\\mathscr{L}(\\mathcal{I})=\\Pi^0_2$ or $\\mathscr{L}(\\mathcal{I})=\\Sigma^0_3$. We conclude with several open questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-16T19:22:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A05",
      "54A20",
      "03E15",
      "03E75",
      "40A05",
      "40A35"
    ],
    "keywords": [
      "ideal limit points",
      "borel complexity",
      "combinatorial properties",
      "open questions",
      "explicit example"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}