{
  "id": "2109.00640",
  "version": "v1",
  "published": "2021-09-01T22:19:14.000Z",
  "updated": "2021-09-01T22:19:14.000Z",
  "title": "Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice",
  "authors": [
    "Kyriakos Keremedis",
    "Eleftherios Tachtsis",
    "Eliza Wajch"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of $\\mathbf{ZF}$, statements: \"Every countable product of compact metrizable spaces is separable (respectively, compact)\" and \"Every countable product of compact metrizable spaces is metrizable\". Statements related to the above-mentioned ones are also studied. Permutation models (among them new ones) are shown in which a countable sum (also a countable product) of metrizable spaces need not be metrizable, countable unions of countable sets are countable and there is a countable family of non-empty sets of size at most $2^{\\aleph_0}$ which does not have a choice function. A new permutation model is constructed in which every uncountable compact metrizable space is of size at least $2^{\\aleph_0}$ but a denumerable family of denumerable sets need not have a multiple choice function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-01T22:19:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E25",
      "03E35",
      "54A35",
      "54E35",
      "54D30",
      "54B10"
    ],
    "keywords": [
      "countable product",
      "countable sum",
      "permutation model",
      "multiple choice function",
      "main aim"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}