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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)

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    $\begingroup$ A partition of unity should do the trick right? $\endgroup$
    – Prahlad Vaidyanathan
    Dec 11, 2013 at 17:02
  • $\begingroup$ That's right! How can I overlook this! Thanks Prahlad!! Just want to make sure - a partition of unity element with support in U means that it vanishes at the boundary of U, right? $\endgroup$
    – Clark Chong
    Dec 11, 2013 at 20:36
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    $\begingroup$ Yes, it would be compactly supported inside $U$ $\endgroup$
    – Prahlad Vaidyanathan
    Dec 12, 2013 at 1:40
  • $\begingroup$ Thanks again! I see that you have an interest in K-theory of operator algebra, would you mind taking a look at my other question: math.stackexchange.com/questions/603371/…? Thanks!! $\endgroup$
    – Clark Chong
    Dec 12, 2013 at 5:40

1 Answer 1

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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 $\operatorname{supp}\phi_V\subset V$.

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