{
  "id": "0910.0439",
  "version": "v1",
  "published": "2009-10-02T16:20:38.000Z",
  "updated": "2009-10-02T16:20:38.000Z",
  "title": "Tychonoff Expansions with Prescribed Resolvability Properties",
  "authors": [
    "W. W. Comfort",
    "Wanjun Hu"
  ],
  "comment": "25 pages, 0 figures",
  "categories": [
    "math.GN"
  ],
  "abstract": "The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and Garc\\'ia-Ferreira (2001): (1) Is every $\\omega$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of ${\\mathcal{KID}}$ expansion, the authors show that {\\it every} suitably restricted Tychonoff topological space $(X,\\sT)$ admits a larger Tychonoff topology (that is, an \"expansion\") witnessing such failure. Specifically the authors show in ZFC that if $(X,\\sT)$ is a maximally resolvable Tychonoff space with $S(X,\\sT)\\leq\\Delta(X,\\sT)=\\kappa$, then $(X,\\sT)$ has Tychonoff expansions $\\sU=\\sU_i$ ($1\\leq i\\leq5$), with $\\Delta(X,\\sU_i)=\\Delta(X,\\sT)$ and $S(X,\\sU_i)\\leq\\Delta(X,\\sU_i)$, such that $(X,\\sU_i)$ is: ($i=1$) $\\omega$-resolvable but not maximally resolvable; ($i=2$) [if $\\kappa'$ is regular, with $S(X,\\sT)\\leq\\kappa'\\leq\\kappa$] $\\tau$-resolvable for all $\\tau<\\kappa'$, but not $\\kappa'$-resolvable; ($i=3$) maximally resolvable, but not extraresolvable; ($i=4$) extraresolvable, but not maximally resolvable; ($i=5$) maximally resolvable and extraresolvable, but not strongly extraresolvable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-02T16:20:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18",
      "03E05",
      "54A10",
      "03E35",
      "54A25",
      "05D05"
    ],
    "keywords": [
      "maximally resolvable",
      "prescribed resolvability properties",
      "tychonoff expansions",
      "restricted tychonoff topological space",
      "tychonoff space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.0439C"
    }
  }
}