{
  "id": "2101.10088",
  "version": "v1",
  "published": "2021-01-25T13:54:33.000Z",
  "updated": "2021-01-25T13:54:33.000Z",
  "title": "Covering versus partitioning with Polish spaces",
  "authors": [
    "Will Brian"
  ],
  "categories": [
    "math.LO",
    "math.GN"
  ],
  "abstract": "Given a completely metrizable space $X$, let $\\mathfrak{par}(X)$ denote the smallest possible size of a partition of $X$ into Polish spaces, and $\\mathfrak{cov}(X)$ the smallest possible size of a covering of $X$ with Polish spaces. Observe that $\\mathfrak{cov}(X) \\leq \\mathfrak{par}(X)$ for every $X$, because every partition of $X$ is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality $\\mathfrak{cov}(X) < \\mathfrak{par}(X)$ can hold for some completely metrizable space $X$. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if $\\mathfrak{cov}(X) < \\mathfrak{par}(X)$ for any completely metrizable $X$, then $0^\\dagger$ exists.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-25T13:54:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polish spaces",
      "strict inequality",
      "metrizable space",
      "huge cardinal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}