Q: Why does a partition of unity has the finite intersection property of compact sets? I have a question regarding partitions of unity. I use the same definition and notation as 2. here (Wikipedia). 
Let $K \subseteq X$ be a compact set. Why do we have that $K \cap \text{supp}\rho_j \neq \varnothing$ for only finitely many $j \in J$? (we have that $\text{supp}\rho_j \subseteq U_i$ as in wikipedia)

A: For each $x\in K$, there exists an open neighbourhood $U_x$ of $x$ where all but finitely many of the functions are $=0$. By compactness, $K$ is covered by finitely many of the $U_x$. All but the finitely many functions that are non-zero in at least one of these finitely many $U_x$ are $=0$ on all of $K$. This does not yet make $\operatorname{supp}(\rho_j)\cap K=\emptyset$, but if we can shrink $U_x$ (replace it with open $V_x$ such that $\overline{V_x}\subseteq U_x$) then all is fine.
