{
  "id": "2305.17059",
  "version": "v1",
  "published": "2023-05-26T16:10:11.000Z",
  "updated": "2023-05-26T16:10:11.000Z",
  "title": "Selective separability properties of Fréchet-Urysohn spaces and their products",
  "authors": [
    "Serhii Bardyla",
    "Fortunato Maesano",
    "Lyubomyr Zdomskyy"
  ],
  "comment": "31 pages; comments are welcome",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "In this paper we study the behaviour of selective separability properties in the class of Frech\\'{e}t-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frech\\'{e}t-Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $\\mathfrak{p}=\\mathfrak{c}$, we construct such an example which is also zero-dimensional and $\\alpha_{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Frech\\'{e}t-Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-26T16:10:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E50",
      "54D65",
      "03E35",
      "54D10",
      "03E17",
      "03E65"
    ],
    "keywords": [
      "selective separability properties",
      "fréchet-urysohn spaces",
      "laver model",
      "zero-dimensional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}