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

TL;DR

This paper presents a method to approximate functions that are continuous off a closed set by functions continuous off a polyhedron, controlling the Hausdorff measure of the singular set, with applications to Lipschitz maps.

Contribution

It introduces a technique to modify the singular set of a continuous map on a simplicial complex, ensuring the new set has controlled Hausdorff measure and preserves continuity properties.

Findings

Existence of a controlled approximation of the singular set.

The approximation preserves Lipschitz continuity if present.

The method applies to arbitrary fine subdivisions of the complex.

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 $K$.