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- W1840998647 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 $K$." @default.
- W1840998647 created "2016-06-24" @default.
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- W1840998647 date "2011-01-11" @default.
- W1840998647 modified "2023-09-27" @default.
- W1840998647 title "On the Approximation of a Function Continuous off a Closed Set by One Continuous Off a Polyhedron" @default.
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