Let $X$ be a locally compact [Hausdorff space](/page/Hausdorff%20Space). Let $C_0(X)$ be the complex $C^*$-[algebra of continuous functions](/theorems/197) $f:X\to\mathbb C$ vanishing at infinity, equipped with pointwise operations and the norm
Let $C_c(X)$ denote the complex [vector space](/page/Vector%20Space) of continuous functions $f:X\to\mathbb C$ whose support $\operatorname{supp}f:=\overline{\{x\in X:f(x)\neq 0\}}$ is compact. For every open subset $U\subset X$, define
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\begin{align*}
I_U:=\{f\in C_0(X): f(x)=0 \text{ for every } x\in X\setminus U\}.
\end{align*}
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Then the assignment from the set $\mathcal{O}(X)$ of open subsets of $X$ to the set of norm-closed two-sided ideals of $C_0(X)$ given by $U\mapsto I_U$ is a bijection.