1. essental ideal

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

Show that $I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ 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 $g\in A$ such that $gI=\{gf:f\in I\}=\{0\}$ so $g=0$ which implies that I is essential. However, this does not show me anything about O.

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

Show that $I=\{f\in A:f(x)=0 \text{ for all }x\notin O\}$ 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 $g\in A$ such that $gI=\{gf:f\in I\}=\{0\}$ so $g=0$ which implies that I is essential. However, this does not show me anything about O.
If $O$ is not dense then there is a nonempty open set $P$ disjoint from $O$. You will need to use something like Urysohn's lemma to get a nonzero function $g\in A$ that vanishes on $O$. Then $fg = 0$ for all $f \in I$.

For the converse, if $O$ is dense in $X$ then any nonzero function $g\in A$ must be nonzero at some point $x_0\in O$. Use Urysohn's lemma again, to show that there is a function $f\in I$ with $f(x_0)\ne0$. That will show that $I$ is essential.