{
  "id": "1805.09708",
  "version": "v1",
  "published": "2018-05-24T14:53:50.000Z",
  "updated": "2018-05-24T14:53:50.000Z",
  "title": "Hausdorff compactifications in ZF",
  "authors": [
    "Kyriakos Keremedis",
    "Eliza Wajch"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "For a compactification $\\alpha X$ of a Tychonoff space $X$, the algebra of all functions $f\\in C(X)$ that are continuously extendable over $% \\alpha X$ is denoted by $C_{\\alpha}(X)$. It is shown that, in a model of $\\textbf{ZF}$, it may happen that a discrete space $X$ can have non-equivalent Hausdorff compactifications $\\alpha X$ and $\\gamma X$ such that $% C_{\\alpha}(X)=C_{\\gamma}(X)$. Amorphous sets are applied to a proof that Glicksberg's theorem that $\\beta X\\times \\beta Y$ is the Cech-Stone compactification of $X\\times Y$ when $X\\times Y$ is a Tychonoff pseudocompact space is false in some models of $\\mathbf{ZF}$. It is noticed that if all Tychonoff compactifications of locally compact spaces had $C^{\\ast}$-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space $X$ to generate a compactification of $X$ are given in $\\mathbf{ZF}$. A concept of a functional \\v{C}ech-Stone compactification is investigated in the absence of the axiom of choice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-24T14:53:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D35",
      "03E25",
      "54D15",
      "03E35"
    ],
    "keywords": [
      "tychonoff space",
      "van douwens choice principle",
      "tychonoff pseudocompact space",
      "non-equivalent hausdorff compactifications",
      "discrete space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}