Prove or disprove: an open set in $\mathbb{R}^2$ is a union of disjoint open balls [duplicate]

Question

I'm trying to prove the following result.

Prove or disprove: every open set in $\mathbb{R}^2$ is a union of countably many disjoint open balls $B_r (x)$, where we allow $r = \infty$.

The use of "disjoint" open balls is throwing me off. If $U \subset \mathbb{R}^2$ is an open set, then for every $u \in U$, I can find an open ball $B_{r_u} (u) \subset U$. Then $U = \bigcup\limits_{u \in U} B_{r_u} (u)$, and I've written $U$ as a union of disjoint open balls. I'm not sure how to "shrink" the radii of the open balls. I could perhaps add a condition that the radii become progressively decreasing, e.g., I start with some radius $r$, then take $\frac{r}{2}$, then $\frac{r}{3}$, and so forth.

I'd appreciate some help on how to show this.

Answer 1

Consider a countable collection $\{B_j\}_{j \ge 1}$ of disjoint open balls in the the open unit square $Q=(0,1)^2$. At most 4 of the points on the boundary of $B_1$ are outside $Q$, and none of the remaining balls $B_2, B_3, \ldots$ can cover any point of $\partial B_1$. Thus $\{B_j\}_{j \ge 1}$ cannot cover $Q$.