{
  "id": "2309.13367",
  "version": "v1",
  "published": "2023-09-23T13:10:43.000Z",
  "updated": "2023-09-23T13:10:43.000Z",
  "title": "Ladder systems and countably metacompact topological spaces",
  "authors": [
    "Rodrigo Carvalho",
    "Tanmay Inamdar",
    "Assaf Rinot"
  ],
  "categories": [
    "math.LO",
    "math.GN"
  ],
  "abstract": "The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\\Delta$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $\\Delta$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $\\kappa$ for which $X_L$ is not countably metacompact, in particular, not a $\\Delta$-space. We prove that an affirmative answer holds for the cardinal $\\kappa=cf(\\beth_{\\omega+1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-23T13:10:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "54G20"
    ],
    "keywords": [
      "countably metacompact topological spaces",
      "ladder system",
      "single cohen real",
      "normal space",
      "unit interval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}