{
  "id": "2101.00513",
  "version": "v1",
  "published": "2021-01-02T20:42:08.000Z",
  "updated": "2021-01-02T20:42:08.000Z",
  "title": "On sequences of homomorphisms into measure algebras and the Efimov Problem",
  "authors": [
    "Piotr Borodulin-Nadzieja",
    "Damian Sobota"
  ],
  "comment": "27 pages",
  "categories": [
    "math.LO",
    "math.GN"
  ],
  "abstract": "For given Boolean algebras $\\mathbb{A}$ and $\\mathbb{B}$ we endow the space $\\mathcal{H}(\\mathbb{A},\\mathbb{B})$ of all Boolean homomorphisms from $\\mathbb{A}$ to $\\mathbb{B}$ with various topologies and study convergence properties of sequences in $\\mathcal{H}(\\mathbb{A},\\mathbb{B})$. We are in particular interested in the situation when $\\mathbb{B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\\mathbb{A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\\mathcal{H}(\\mathbb{A},\\mathbb{B})$ for a Boolean algebra $\\mathbb{B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\\mathbb{A}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-02T20:42:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measure algebra",
      "efimov problem",
      "boolean algebra",
      "study convergence properties",
      "boolean homomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}