It is well-known that any abelian C*-algebra can be identified, by Gelfand transform, as $C_0(\Omega)$. Here, the spectrum $\Omega$ is locally compact Hausdorff in general, and compact exactly when the algebra is unital. In this survey article, we collect some results about the relation between the metrizability of $\Omega$ and the separability of $C_0(\Omega)$. A brief discussion for the topological structure of the spectrum of a general separable C*-algebra $A$ is also given.