Big Rudin 1.40: Open Set is a countable union of closed disks?

An arbitrary open set of $\mathbb{C}$ is a countable union of open balls - specifically, if $U\subseteq \mathbb{C}$ is open, then $U$ is the union of the "rational" open balls contained in $U$, that is, those open balls whose centers are complex numbers whose real and imaginary parts are rational, and whose radii are rational (since $\mathbb{Q}$ is countable, there can only be countably many such open balls). Now note that any open ball is a countable union of closed balls - specifically, $\{z\in\mathbb{C}:|z-\alpha|<\epsilon\}$ is equal to the union $$\bigcup_{n\in\mathbb{N}}\{z\in\mathbb{C}:|z-\alpha|\leq\epsilon-\tfrac{1}{n}\}.$$ Lastly, note that a countable union of countable unions is still a countable union.