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**Mauritzvdworm** Let $\displaystyle A=C_0(X)$ where X is some locally compact Hausdorff space and let $\displaystyle O\subset X$ be open.

Show that $\displaystyle I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ is an essential ideal in A if and only if O is dense in X

It is easy to see that I is an ideal in A

My plan involved choosing a specific $\displaystyle g\in A$ such that $\displaystyle gI=\{gf:f\in I\}=\{0\}$ so $\displaystyle g=0$ which implies that I is essential. However, this does not show me anything about O.