IJPAM: Volume 68, No. 3 (2011)

APPROXIMATING A FUNCTION CONTINUOUS OFF
A CLOSED SET BY ONE CONTINUOUS OFF A POLYHEDRON

Steven P. Ellis
Department of Psychiatry
College of Physicians and Surgeons
Columbia University
1051, Riverside Drive, P.O. Box 42, New York, NY 10032, USA
e-mail: spe4@columbia.edu


Abstract. Let $P$ be a finite simplicial complex (i.e., a finite collection of simplices that fit together nicely) with underlying space (union of simplices in $P$) $\vert P\vert$. Let $Q$ be a subcomplex of $P$. Let $a \geq 0$. Then there exists $K < \infty$, depending only on $a$ and $Q$, with the following property. Let $\Ss \subset \vert P\vert$ be closed and suppose $\Phi$ is a continuous map of $\vert P\vert \setminus \Ss$ into some topological space $\F$ (``$\setminus$'' indicates set-theoretic subtraction). Suppose $\dim (\tilde{\Ss} \cap \vert Q\vert) \leq a$, where ``$\dim$'' indicates Hausdorff dimension. Then there exists $\tilde{\Ss} \subset \vert P\vert$ such that $\tilde{\Ss} \cap \vert Q\vert$ is the underlying space of a subcomplex of $Q$ and there is a continuous map $\tilde{\Phi}$ of $\vert P\vert \setminus \tilde{\Ss}$ into $\F$ such that:

  • $\Hm^{a} \bigl( \tilde{\Ss} \cap \vert Q\vert \bigr) \leq K \Hm^{a} \bigl( \Ss \cap \vert Q\vert \bigr)$, where $\Hm^{a}$ denotes $a$-dimensional Hausdorff measure;
  • if $x \in \tilde{\Ss}$ then $x$ belongs to a simplex in $P$ intersecting $\Ss$;
  • if $x \in \vert P\vert \setminus \Ss$, $x \in \sigma \in P$, and $\sigma$ does not intersect any simplex in $Q$ whose simplicial interior intersects $\Ss$, then $\tilde{\Phi}(x)$ is defined and equals $= \Phi(x)$;
  • if $\sigma \in P$ then $\tilde{\Phi}(\sigma \setminus \tilde{\Ss}) \subset \Phi(\sigma \setminus \Ss)$;
  • if $\F$ is a metric space and $\Phi$ is locally Lipschitz on $\vert P\vert \setminus \Ss$ then $\tilde{\Phi}$ is locally Lipschitz on $\vert P\vert \setminus \tilde{\Ss}$; and
  • $\dim (\tilde{\Ss} \cap \vert Q\vert) \leq \dim (\Ss \cap \vert Q\vert)$ and $\dim \tilde{\Ss} \leq \dim \Ss$.

Moreover, $P$ can be replaced by an arbitrarily fine subdivision without changing $K$. Consequently, modulo subdivision, if $\epsilon > 0$, we may assume $\tilde{\Phi}(x) = \Phi(x)$ if $\text{\rm dist}\,(x, \Ss) > \epsilon$ and we may assume $\max \bigl\{ \text{\rm dist}\, ( y, \Ss) : y \in \tilde{\Ss} \bigr\} < \epsilon$.

Note that $\Ss$ can be any closed subset of $\vert P\vert$. For example, no rectifiability assumptions on $\Ss$ are required.



Received: December 23, 2010

AMS Subject Classification: 28A75, 51M20

Key Words and Phrases: simplicial complex, deformation theorem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 68
Issue: 3