{
  "id": "2311.10209",
  "version": "v1",
  "published": "2023-11-16T21:54:59.000Z",
  "updated": "2023-11-16T21:54:59.000Z",
  "title": "Cardinal invariants of a meager ideal",
  "authors": [
    "Will Brian"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $\\mathcal M_X$ denote the ideal of meager subsets of a topological space $X$. We prove that if $X$ is a completely metrizable space without isolated points, then the smallest cardinality of a non-meager subset of $X$, denoted $\\mathrm{non}(\\mathcal M_X)$, is exactly $\\mathrm{non}(\\mathcal M_X) = \\mathrm{cf}[\\kappa]^\\omega \\cdot \\mathrm{non}(\\mathcal M_{\\mathbb R})$, where $\\kappa$ is the minimum weight of a nonempty open subset of $X$. We also characterize the additivity and covering numbers for $\\mathcal M_X$ in terms of simple topological properties of $X$. Some bounds are proved and some questions raised concerning the cofinality of $\\mathcal M_X$ and the cofinality of the related ideal of nowhere dense subsets of $X$. We also show that if $X$ is a compact Hausdorff space with $\\pi$-weight $\\kappa$, then $\\mathrm{non}(\\mathcal M_X) \\leq \\mathrm{cf}[\\kappa]^\\omega \\cdot \\mathrm{non}(\\mathcal M_{\\mathbb R})$. This bound for compact Hausdorff spaces is not sharp, in the sense that it is consistent for such a space to have non-meager subsets of even smaller cardinality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-16T21:54:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cardinal invariants",
      "meager ideal",
      "compact hausdorff space",
      "non-meager subset",
      "nonempty open subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}