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

Let $P$ be a finite simplicial comple with underlying space (union of simplices in $P$) $\mathrm{\mid }P\mathrm{\mid }$. Let $Q$ be a subcomplex of $P$. Let $a\ge 0$. Then there exists $K<\mathrm{\infty }$, \emph{depending only on $a$ and $Q$,} with the following property. Let $\mathcal{S}\subset \mathrm{\mid }P\mathrm{\mid }$ be closed and suppose $\mathrm{\Phi }$ is a continuous map of $\mathrm{\mid }P\mathrm{\mid }\setminus \mathcal{S}$ into some topological space $\mathcal{F}$. Suppose $\dim (\tilde{\mathcal{S}}\cap \mathrm{\mid }Q\mathrm{\mid })\le a$, where “$\dim$” = Hausdorff dimension. Then there exists $\tilde{$\mathcal{S}$} \subset |P|$suchthat$\tilde{$\mathcal{S}$} \cap |Q|$ is the underlying space of a subcomplex of $Q$ and there is a continuous map $\tilde{\mathrm{\Phi }}$ of $|P| \setminus \tilde{$\mathcal{S}$}$into$$\mathcal{F}$$suchthat$$\mathcal{H}$^{a} \bigl(\tilde{$\mathcal{S}$} \cap |Q| \bigr) \leq K $\mathcal{H}$^{a} \bigl($\mathcal{S}$ \cap |Q| \bigr)$,where$$\mathcal{H}$^{a}$denotes$a$-dimensionalHausdorffmeasure;if$x \in \tilde{$\mathcal{S}$}$then$x$belongstoasimplexin$P$intersecting$$\mathcal{S}$$;if$x \in |P| \setminus $\mathcal{S}$$,$x \in \sigma \in P$,and$\sigma$ does not intersect any simplex in $Q$ whose simplicial interior intersects $\mathcal{S}$, then $\tilde{\mathrm{\Phi }}(x)$ is defined and equals $=\mathrm{\Phi }(x)$; if $\sigma \in P$ then $\tilde{\Phi}(\sigma \setminus \tilde{$\mathcal{S}$}) \subset \Phi(\sigma \setminus $\mathcal{S}$)$;andif$$\mathcal{F}$$isametricspaceand$\Phi$ is locally Lipschitz on $\mathrm{\mid }P\mathrm{\mid }\setminus \mathcal{S}$ then $\tilde{\mathrm{\Phi }}$ is locally Lipschitz on $\mathrm{\mid }P\mathrm{\mid }\setminus \tilde{\mathcal{S}}$ Moreover, $P$ can be replaced by an arbitrarily fine subdivision without changing $K$.