{
  "id": "1809.06819",
  "version": "v1",
  "published": "2018-09-18T16:33:22.000Z",
  "updated": "2018-09-18T16:33:22.000Z",
  "title": "Countable dense homogeneity and $λ$-sets",
  "authors": [
    "Rodrigo Hernández-Gutiérrez",
    "Michael Hrušák",
    "Jan van Mill"
  ],
  "journal": "Fund. Math. 226 (2014) 157-172",
  "categories": [
    "math.GN"
  ],
  "abstract": "We show that all sufficiently nice $\\lambda$-sets are countable dense homogeneous ($\\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\\kappa \\le \\mathfrak{b}$ there is a countable dense homogeneous metric space of size $\\kappa$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $\\kappa$ is equivalent to the existence of a $\\lambda$-set of size $\\kappa$. On the other hand, it is consistent with the continuum arbitrarily large that every $\\mathsf{CDH}$ metric space has size either $\\omega_1$ or size $\\mathfrak c$. An example of a Baire $\\mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $\\mathsf{CDH}$ space in ZFC.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-18T16:33:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H05",
      "03E15",
      "54E50"
    ],
    "keywords": [
      "countable dense homogeneity",
      "countable dense homogeneous metric space",
      "continuum arbitrarily large",
      "van mill",
      "consistent"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}