Show that every open subset can be represented as a union of at most countably many balls Consider $\mathbb{R^n}$ with the $\infty$-norm.
Show that every open subset can be represented as a union of at most countably many balls.
This might have been answered before but I do not understand the answers on there.
I have an idea that it needs to use the fact that the rationals are dense but I'm not sure how to make it rigorous.
I want to avoid using metric space notation but balls are fine.

Let $U$ be open in $\mathbb R^n.$ For every $x\in U,$ there exists $q_x\in \mathbb Q^n$ and $r_x\in \mathbb Q$ such that $x\in B(q_x,r_x)\subset U.$ Therefore
$$U = \cup_{x\in U} \{x\} \subset \cup_{x\in U} B(q_x,r_x) \subset U.$$
Therefore $U = \cup_{x\in U} B(q_x,r_x),$ and there are at most countably many distinct $B(q_x,r_x).$
