TITLE: Does not exist cover of $\mathbb{R}^n$ by disjoint closed balls
QUESTION [6 upvotes]: Does not exist cover of $\mathbb{R}^n$ by disjoint closed balls with positive radius.

My attempt: Suppose that exists, we can write:
$\mathbb{R}^n=\displaystyle\bigcup_{i=1}^{\infty} B_{i}$. Let $C$ denote the set of center points of these balls. All points of $C$ are isolated so $C$ is countable. Associating each point of $C$ to the ball of wich this point is the center,it is easy see that this association is a bijection, concluding that set of balls is countable.
The set of limit points of $C$, $C^{'}=\displaystyle\bigcup_{i=1}^{\infty} \partial B_{i}$. I thought it would be easy to find a contradiction there, I was wrong. Until now, I don't use hypothesis that $B_{i}$ is closed, and I think that is it that lack in my demostration.
Is it true if the balls are open?
Thank you for any help.

REPLY [8 votes]: For the Baire category proof.
Use the OP proof up to
$$
C^{'}=\displaystyle\bigcup_{i=1}^{\infty} \partial B_{i} .
$$
Then note $C^{'}$ is a complete metric space, written as a countable union of sets $\partial B_i$, which are closed (in $C'$) sets with empty interior (in $C'$).