On the Approximation of a Function Continuous off a Closed Set by One Continuous Off a Polyhedron

zbMATH Open1216.28005arXiv1101.2184MaRDI QIDQ3004638

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Publication date: 3 June 2011

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 $ageq0$. Then there exists $K<infty$, emph{depending only on $a$ and $Q$,} with the following property. Let $mathcalSsubset|P|$ be closed and suppose $Phi$ is a continuous map of $|P|setminusmathcalS$ into some topological space $mathcalF$. Suppose $dim(ildemathcalScap|Q|)leqa$, where "$dim$" = Hausdorff dimension. Then there exists $ildemathcalSsubset|P|$ such that $ildemathcalScap|Q|$ is the underlying space of a subcomplex of $Q$ and there is a continuous map $ildePhi$ of $|P|setminusildemathcalS$ into $mathcalF$ such that $$, where $mathcal{H}^a$ denotes $a$-dimensional Hausdorff measure; if $xinildemathcalS$ then $x$ belongs to a simplex in $P$ intersecting $mathcalS$; if $xin|P|setminusmathcalS$, $xinsigmainP$, and $sigma$ does not intersect any simplex in $Q$ whose simplicial interior intersects $mathcalS$, then $ildePhi(x)$ is defined and equals $=Phi(x)$; if $sigmainP$ then $ildePhi(sigmasetminusildemathcalS)subsetPhi(sigmasetminusmathcalS)$; and if $mathcalF$ is a metric space and $Phi$ is locally Lipschitz on $|P|setminusmathcalS$ then $ildePhi$ is locally Lipschitz on $|P|setminusildemathcalS$ Moreover, $P$ can be replaced by an arbitrarily fine subdivision without changing $K$.

Full work available at URL: https://arxiv.org/abs/1101.2184

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