Locally finite smooth partitions of unity on Euclidean open sets

If $\mathrm{\Omega }=\mathrm{\varnothing }$, the empty family proves the claim. If $n=0$ and $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, then $\mathrm{\Omega }={\mathbb{R}}^0$ is a singleton; choose one cover member containing it and the single function $\eta =1$, which is smooth and compactly supported. Thus assume $n\ge 1$ and $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$. For $m\ge 1$ set $K_m=\{x:\mathrm{\mid }x\mathrm{\mid }\le m,\mathrm{dist}(x,{\mathrm{\Omega }}^c)\ge 1\mathrm{/}m\}$ when $\mathrm{\Omega }\mathrm{\ne }{\mathbb{R}}^n$, and set $K_m=\bar{B}(0,m)$ when $\mathrm{\Omega }={\mathbb{R}}^n$; put $K_m=\mathrm{\varnothing }$ for $m\le 0$. Each $K_m$ is compact by [F1], the sets are nested, $K_m\subset \mathrm{int}{K}_{m+1}$, and their interiors cover $\mathrm{\Omega }$. The compact shells $S_m:=K_m\setminus \mathrm{int}{K}_{m-1}$ cover $\mathrm{\Omega }$ and lie in the open sets $U_m:=\mathrm{int}{K}_{m+1}\setminus K_{m-2}$. The family $(U_m)$ is locally finite: any compact subset of $\mathrm{\Omega }$ lies in some $\mathrm{int}{K}_N$ and hence in $K_{m-2}$ for all sufficiently large $m$.

Fix once a smooth nonnegative cutoff $\theta$ equal to one on $\bar{B}(0,1)$ and supported in $B(0,2)$, whose existence is [F3]. In each $U_m$, consider the rational inner balls $B(q,r)$ for which $\bar{B}(q,2r)\subset U_m\cap V_{\mathrm{\ell }}$ for some cover index $\mathrm{\ell }$. Indexed by $(\mathrm{\ell },q,r)$, these balls form a countable cover of $S_m$: for each point choose a cover member containing it, then density [F2] gives a rational center and radius with the required doubled closure. Compactness of $S_m$ gives a finite subcover. Order all finite lists of eligible indexed balls by natural-number codes and take the first list covering $S_m$; thus no choice principle is needed. For each selected ball use the fixed rescaled cutoff ${\psi }_{m,j}(x):=\theta ((x-q)\mathrm{/}r)$. The family is locally finite because each shell contributes finitely many cutoffs supported in $U_m$, and each support also lies in its assigned cover member. Its sum $s:={\sum }_{m,j}{\psi }_{m,j}$ is smooth and positive on $\mathrm{\Omega }$ because the shells cover $\mathrm{\Omega }$.

Define ${\eta }_{m,j}:={\psi }_{m,j}\mathrm{/}s$. Then each function is smooth, nonnegative and compactly supported in $\mathrm{\Omega }$, the family is locally finite, and ${\sum }_{m,j}{\eta }_{m,j}=1$. Each support remains in its assigned cover member, and enumeration gives the required countable family.