A finite compactly supported partition of unity near a compact set

Consider all triples $(V,W,i)$ with $V,W$ open, $1\le i\le n$, [given, L1] and $\bar{V}\subseteq W\subseteq \bar{W}\subseteq U_i,$ where the displayed closures are compact. Local compactness and the Hausdorff property show that the sets $V$ occurring in these triples cover $K$. Compactness therefore gives finitely many triples $(V_j,W_j,i_j)$ whose $V_j$ cover $K$. Apply [L1] to $\overline{V_j}\subseteq W_j$ to obtain $h_j\in C_c(X)$ with $0\le h_j\le 1$, $h_j=1$ on $\overline{V_j}$, and $h_j=0$ off $W_j$. Consequently $\mathrm{supp}{h}_j\subseteq \overline{W_j}\subseteq U_{i_j}$.

Put $h={\sum }_j{h}_j$. Then $h\ge 1$ on $K$, so $\{h>1\mathrm{/}2\}$ is an [step 1.1, L1, choose] open neighbourhood of $K$. Choose open sets $O,W$ with $K\subseteq O\subseteq \bar{O}\subseteq W\subseteq \bar{W}\subseteq \{h>1\mathrm{/}2\},$ where the displayed closures are compact. Apply [L1] to $\bar{O}\subseteq W$ to obtain $g\in C_c(X)$ with $0\le g\le 1$, $g=1$ on $\bar{O}$, and $g=0$ off $W$. Thus $\mathrm{supp}g\subseteq \bar{W}\subseteq \{h>1\mathrm{/}2\}$, and $g=1$ on the open neighbourhood $O$ of $K$.

For $1\le i\le n$, set ${\phi }_i(x)=\{\begin{array}{ll}g(x){\displaystyle \frac{\sum _{j:i_j=i}{h}_j(x)}{h(x)},}& h(x)>0,\\ 0,& h(x)=0.\end{array}$ The quotient is only used on $\mathrm{supp}g\subseteq \{h>1\mathrm{/}2\}$, so extension by zero is continuous. Each ${\phi }_i$ is nonnegative, compactly supported in $U_i$, and ${\sum }_i{\phi }_i=g$, hence the sum is $1$ on $O$.