{
  "id": "1509.01420",
  "version": "v1",
  "published": "2015-09-04T11:57:23.000Z",
  "updated": "2015-09-04T11:57:23.000Z",
  "title": "Anti-Urysohn spaces",
  "authors": [
    "István Juhász",
    "Lajos Soukup",
    "Zoltán Szentmiklóssy"
  ],
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "All spaces are assumed to be infinite Hausdorff spaces. We call a space \"anti-Urysohn\" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\\bullet$ for every infinite cardinal ${\\kappa}$ there is a space of size ${\\kappa}$ in which fewer than $cf({\\kappa})$ many non-empty regular closed sets always intersect; $\\bullet$ there is a locally countable AU space of size $\\kappa$ iff $\\omega \\le \\kappa \\le 2^{\\mathfrak c}$. A space with at least two non-isolated points is called \"strongly anti-Urysohn\" $($SAU in short$)$ iff any two infinite closed sets in it intersect. We prove that $\\bullet$ if $X$ is any SAU space then $ \\mathfrak s\\le |X|\\le 2^{2^{\\mathfrak c}}$; $\\bullet$ if $\\mathfrak r=\\mathfrak c$ then there is a separable, crowded, locally countable, SAU space of cardinality $\\mathfrak c$; \\item if $\\lambda > \\omega$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $\\kappa$ for all $\\kappa \\in [\\omega_1,\\lambda]$; $\\bullet$ if GCH holds and $\\kappa \\le\\lambda$ are uncountable regular cardinals then in some CCC generic extension we have $\\mathfrak s={\\kappa}$, $\\,\\mathfrak c={\\lambda}$, and for every cardinal ${\\mu}\\in [\\mathfrak s, \\mathfrak c]$ there is an SAU space of cardinality ${\\mu}$. The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality $> \\mathfrak c$ can exist remain open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-04T11:57:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A25",
      "54A35",
      "54D10",
      "03E04"
    ],
    "keywords": [
      "sau space",
      "anti-urysohn spaces",
      "ccc generic extension",
      "non-empty regular closed sets",
      "non-emty regular closed sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150901420J"
    }
  }
}