Let $\mathcat{Haus}$ be the category of locally compact Hausdorff spaces with continuous proper maps as morphisms. And, let $\mathcat{C^*Alg}$ be the category of commutative $C^*$ -algebras with proper *-homomorphisms (send approximate units into approximate units) as morphisms. There is a contravariant functor $C\colon \mathcat{Haus}^\op \to \mathcat{C^*Alg}$ which sends each locally compact Hausdorff space $X$ to the commutative $C^*$ -algebra $C_0(X)$ ($C(X)$ if $X$ is compact). Conversely, there is a contravariant functor $M\colon \mathcat{C^*Alg}^\op \to \mathcat{Haus}$ which sends each commutative $C^*$ -algebra $A$ to the space of characters on $A$ (with the Gelfand topology).

The functors $C$ and $M$ are an equivalence of categories.