{
  "id": "2310.17984",
  "version": "v1",
  "published": "2023-10-27T08:55:40.000Z",
  "updated": "2023-10-27T08:55:40.000Z",
  "title": "Countable spaces, realcompactness, and the pseudointersection number",
  "authors": [
    "Claudio Agostini",
    "Andrea Medini",
    "Lyubomyr Zdomskyy"
  ],
  "comment": "16 pages",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\\mathsf{Exp}(X)$ the smallest infinite cardinal $\\kappa$ such that $X$ is homeomorphic to a closed subspace of $\\mathbb{R}^\\kappa$. Our main result shows that, given a cardinal $\\kappa$, the following conditions are equivalent: $(1)$ There exists a countable crowded space $X$ such that $\\mathsf{Exp}(X)=\\kappa$, $(2)$ $\\mathfrak{p}\\leq\\kappa\\leq\\mathfrak{c}$. In fact, in the case $\\mathfrak{d}\\leq\\kappa\\leq\\mathfrak{c}$, every countable dense subspace of $2^\\kappa$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\\kappa$ has pseudocharacter at most $\\kappa$ in any compactification. This will allow us to calculate $\\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-27T08:55:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D60",
      "03E17",
      "54G12"
    ],
    "keywords": [
      "countable space",
      "pseudointersection number",
      "realcompactness",
      "smallest infinite cardinal",
      "pseudocharacter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}