{
  "id": "1101.2184",
  "version": "v3",
  "published": "2011-01-11T19:27:24.000Z",
  "updated": "2011-05-03T17:26:03.000Z",
  "title": "On the Approximation of a Function Continuous off a Closed Set by One Continuous Off a Polyhedron",
  "authors": [
    "Steven P. Ellis"
  ],
  "comment": "75 pages, 1 Postscript figure, packages: amssymb, latexsym, amscd, epsfig. A shorter version of this paper will appear in International Journal of Pure and Applied Mathematics",
  "categories": [
    "math.GN",
    "math.MG"
  ],
  "abstract": "Let $P$ be a finite simplicial comple with underlying space (union of simplices in $P$) $|P|$. Let $Q$ be a subcomplex of $P$. Let $a \\geq 0$. Then there exists $K < \\infty$, \\emph{depending only on $a$ and $Q$,} with the following property. Let $\\mathcal{S} \\subset |P|$ be closed and suppose $\\Phi$ is a continuous map of $|P| \\setminus \\mathcal{S}$ into some topological space $\\mathcal{F}$. Suppose $\\dim (\\tilde{\\mathcal{S}} \\cap |Q|) \\leq a$, where \"$\\dim$\" = Hausdorff dimension. Then there exists $\\tilde{\\mathcal{S}} \\subset |P|$ such that $\\tilde{\\mathcal{S}} \\cap |Q|$ is the underlying space of a subcomplex of $Q$ and there is a continuous map $\\tilde{\\Phi}$ of $|P| \\setminus \\tilde{\\mathcal{S}}$ into $\\mathcal{F}$ such that $\\mathcal{H}^{a} \\bigl(\\tilde{\\mathcal{S}} \\cap |Q| \\bigr) \\leq K \\mathcal{H}^{a} \\bigl(\\mathcal{S} \\cap |Q| \\bigr)$, where $\\mathcal{H}^{a}$ denotes $a$-dimensional Hausdorff measure; if $x \\in \\tilde{\\mathcal{S}}$ then $x$ belongs to a simplex in $P$ intersecting $\\mathcal{S}$; if $x \\in |P| \\setminus \\mathcal{S}$, $x \\in \\sigma \\in P$, and $\\sigma$ does not intersect any simplex in $Q$ whose simplicial interior intersects $\\mathcal{S}$, then $\\tilde{\\Phi}(x)$ is defined and equals $= \\Phi(x)$; if $\\sigma \\in P$ then $\\tilde{\\Phi}(\\sigma \\setminus \\tilde{\\mathcal{S}}) \\subset \\Phi(\\sigma \\setminus \\mathcal{S})$; and if $\\mathcal{F}$ is a metric space and $\\Phi$ is locally Lipschitz on $|P| \\setminus \\mathcal{S}$ then $\\tilde{\\Phi}$ is locally Lipschitz on $|P| \\setminus \\tilde{\\mathcal{S}}$ Moreover, $P$ can be replaced by an arbitrarily fine subdivision without changing $K$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-05-03T17:26:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "51M20"
    ],
    "keywords": [
      "closed set",
      "function continuous",
      "approximation",
      "polyhedron",
      "dimensional hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 75,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.2184E"
    }
  }
}