Finite smooth cutoffs relative to compact sets and boundary neighborhoods

If $K=\mathrm{\varnothing }$, take $\chi =0$ and take every function indexed by the prescribed finite coordinate cover to be zero; for an empty cover use the empty family. The required equalities near $K$ hold on the empty open neighborhood, and every support is empty. Henceforth suppose $K\mathrm{\ne }\mathrm{\varnothing }$.

For an open neighborhood $W$ of a point $p$, take a boundary chart $\phi :U\to O\subseteq {\mathbb{H}}^n$ and write $a=\phi (p)$. In positive dimension choose $0<r<s$ with ${\bar{B}}_s(a)\cap {\mathbb{H}}^n\subseteq \phi (U\cap W)$, using relative openness. By [F2] take a translated Euclidean bump equal to one on ${\bar{B}}_r(a)$ and supported in $B_s(a)$. Restrict it to $O$ and pull it back to $U$. The set $C={\phi }^{-1}({\bar{B}}_s(a)\cap {\mathbb{H}}^n)$ is compact by [F4], closed in $R$, and contained in $U\cap W$. The pullback is zero on $U\setminus C$, so extension by zero on $R\setminus C$ is smooth: the two formulas agree on their open overlap, and the boundary-chart formula has a smooth ambient extension. Its support is closed in $R$, is therefore compact by [F4], lies in $C\subseteq W$, and it equals one on the open neighborhood ${\phi }^{-1}(B_r(a)\cap {\mathbb{H}}^n)$ of $p$. In dimension zero, a singleton chart is open and closed, and its indicator is a smooth bump with that singleton as compact support.

Consider all the bumps and inner neighborhoods constructed in step 1.2 with $p\in K$ and $W=V$. Their inner neighborhoods form an indexed open cover of $K$; [F3] gives finitely many, with bumps $b_1,\dots ,b_N$. Set $\chi =1-{\prod }_{a=1}^N(1-b_a)$. This is smooth by the iterated product rule, takes values in $[0,1]$, and equals one on the union of the inner neighborhoods, since there one factor is zero. Its support lies in the finite union of the closed supports of the $b_a$, which is closed in $R$ and contained in $V$. Both this finite union and $\mathrm{supp}\chi$ are closed in compact Hausdorff $R$, hence compact by [F4]. If $K\subset R^{\circ }$, perform the same construction with $W=V\cap R^{\circ }$; the resulting support is compactly contained in $R^{\circ }$. This proves the first assertion and its interior refinement without taking a minimum or using a scalar cutoff.

For the second assertion, write the prescribed coordinate neighborhoods as $U_1,\dots ,U_m$. Use all the local constructions of step 1.2 with $p\in K\cap U_j$ and $W=U_j\cap R^{\circ }$, retaining the index $j$. Their inner neighborhoods cover $K$, so [F3] gives finitely many bumps $b_a$, each with compact support in its assigned $U_{j(a)}\cap R^{\circ }$. Set $S={\sum }_a{b}_a$ and $W_0=\{S>0\}\cap R^{\circ }$. This is open and contains $K$. Apply the already proved construction in step 2.1 with $V=W_0$ to obtain a smooth $\psi :R\to [0,1]$ equal to one near $K$ with compact support contained in $W_0$.

For $1\le j\le m$, define ${\chi }_j=\psi \left({\sum }_{a:j(a)=j}{b}_a\right)\mathrm{/}S$ on $W_0$, and set it to zero elsewhere; an empty numerator sum is zero. These functions are smooth on $W_0$ by [F5]. At every point outside $W_0$, the complement of the closed set $\mathrm{supp}\psi \subset W_0$ is a neighborhood on which both formulas are zero, proving smooth extension. Each ${\chi }_j$ is nonnegative and at most one. Its support lies in the intersection of $\mathrm{supp}\psi$ with the finite union of the supports assigned to $j$. This support is closed in $R$, so [F4] makes it compact; it is therefore compactly contained in $U_j\cap R^{\circ }$. Summing the formulas gives ${\sum }_j{\chi }_j=\psi$ on $W_0$, and both sides vanish off $W_0$. Thus the sum equals one near $K$, proving the second assertion.