Test function cutoffs and euclidean localization

For each point of $K$, there is a rational center $q$ and positive rational $r$ such that the point belongs to $B(q,r)$ and $\bar{B}(q,2r)\subseteq \mathrm{\Omega }$. These inner balls cover $K$, so a finite list suffices by compactness. Put $b_j(x)=b((x-q_j)\mathrm{/}{r}_j)$ and $\chi =1-{\prod }_j(1-b_j)$. Then $\chi$ is smooth, lies in $[0,1]$, equals one on the union of the inner balls, and has support in the finite union of the compact outer balls inside $\mathrm{\Omega }$. For $K=\mathrm{\varnothing }$ take the empty product and $\chi =0$.

For nonempty $\mathrm{\Omega }$, set $K_j=\{x:\mathrm{\mid }x\mathrm{\mid }\le j,\mathrm{dist}(x,{\mathbb{R}}^n\setminus \mathrm{\Omega })\ge 1\mathrm{/}j\}$ for $j\ge 1$, interpreting distance to the empty set as infinity; set $K_0=K_{-1}=\mathrm{\varnothing }$. These are compact subsets of $\mathrm{\Omega }$, $K_j\subseteq \mathrm{int}{K}_{j+1}$, and their interiors cover $\mathrm{\Omega }$. Closedness follows from continuity of distance (its absolute difference is at most the distance of the two points), and boundedness gives compactness. The shell $H_j=K_j\setminus \mathrm{int}{K}_{j-1}$ is compact and lies in the open set $G_j=\mathrm{int}{K}_{j+1}\setminus K_{j-2}$.

Fix an enumeration of rational center/radius pairs and a coding of finite lists by natural numbers. For each $j$, consider pairs with $\bar{B}(q,2r)\subseteq G_j$ and with this closed ball contained in some $U\in \mathcal{U}$. Their inner balls cover $H_j$: at any shell point openness of $G_j$ and of one cover member gives a sufficiently small ball, then a rational center and radius. Compactness gives a finite subcover. Select the least code of a finite list that covers $H_j$, taking the empty list for an empty shell. This is a specified function of $j$, not Countable Choice.

Form the corresponding translated dilates $b_{j,k}$ of F1. Their supports lie in $G_j$, and each support lies in some cover member. This family is locally finite: a point has a neighborhood inside some $K_N$, and supports with $j\ge N+2$ miss $K_N$, while only finitely many balls occur at each of the finitely many earlier stages. Every point belongs to some shell, so $s={\sum }_{j,k}{b}_{j,k}$ is everywhere positive. The sum is locally finite and smooth; hence ${\eta }_{j,k}=b_{j,k}\mathrm{/}s$ are smooth, nonnegative, have compact support in the same outer balls, and sum to one. The double index is countable. For empty $\mathrm{\Omega }$ use the empty family. These constructions choose no cover labels and no arbitrary sequence of witnesses. $\mathrm{\square }$