{
  "id": "1008.5371",
  "version": "v2",
  "published": "2010-08-31T17:21:36.000Z",
  "updated": "2011-08-15T18:37:46.000Z",
  "title": "Exactly $n$-resolvable Topological Expansions",
  "authors": [
    "W. W. Comfort",
    "Wanjun Hu"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "For $\\kappa$ a cardinal, a space $X=(X,\\sT)$ is $\\kappa$-{\\it resolvable} if $X$ admits $\\kappa$-many pairwise disjoint $\\sT$-dense subsets; $(X,\\sT)$ is {\\it exactly} $\\kappa$-{\\it resolvable} if it is $\\kappa$-resolvable but not $\\kappa^+$-resolvable. The present paper complements and supplements the authors' earlier work, which showed for suitably restricted spaces $(X,\\sT)$ and cardinals $\\kappa\\geq\\lambda\\geq\\omega$ that $(X,\\sT)$, if $\\kappa$-resolvable, admits an expansion $\\sU\\supseteq\\sT$, with $(X,\\sU)$ Tychonoff if $(X,\\sT)$ is Tychonoff, such that $(X,\\sU)$ is $\\mu$-resolvable for all $\\mu<\\lambda$ but is not $\\lambda$-resolvable (cf. Theorem~3.3 of \\cite{comfhu10}). Here the \"finite case\" is addressed. The authors show in ZFC for $1<n<\\omega$: (a) every $n$-resolvable space $(X,\\sT)$ admits an exactly $n$-resolvable expansion $\\sU\\supseteq\\sT$; (b) in some cases, even with $(X,\\sT)$ Tychonoff, no choice of $\\sU$ is available such that $(X,\\sU)$ is quasi-regular; (c) if $n$-resolvable, $(X,\\sT)$ admits an exactly $n$-resolvable quasi-regular expansion $\\sU$ if and only if either $(X,\\sT)$ is itself exactly $n$-resolvable and quasi-regular or $(X,\\sT)$ has a subspace which is either $n$-resolvable and nowhere dense or is $(2n)$-resolvable. In particular, every $\\omega$-resolvable quasi-regular space admits an exactly $n$-resolvable quasi-regular expansion. Further, for many familiar topological properties $\\PP$, one may choose $\\sU$ so that $(X,\\sU)\\in\\PP$ if $(X,\\sT)\\in\\PP$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-15T18:37:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18",
      "03E05",
      "54A10"
    ],
    "keywords": [
      "resolvable topological expansions",
      "resolvable quasi-regular expansion",
      "resolvable quasi-regular space admits",
      "finite case",
      "earlier work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.5371C"
    }
  }
}