If $U$ and $V$ are two open subsets of $\mathbb{C}$, is it true that $C_0(U)+C_0(V)$ is dense in $C_0(U \cup V)$?
Or better yet, is $C_0(U)+C_0(V) = C_0(U \cup V)$?
If it were true, I am wondering if we may replace $\mathbb{C}$ by $X$, a locally compact Hausdorff space. (the original motivation of the question is for $X$ the maximal ideal space of a commutative Banach space)
Expanding a comment by Prahlad Vaidyanathan: A continuous partition of unity subordinate to a finite cover (such as $U,V$) exists for every locally compact Hausdorff space. Thus, for every $f\in C_0(U\cup V)$ one can write $f=f\phi_U+f\phi_V$ where $\operatorname{supp}\phi_U\subset U$ and $\operatorname{supp}\phi_V\subset V$.