For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

Question

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed

balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda \in \Gamma} D_{\lambda}$ is path connected ? ( I am also allowing points to be closed balls , of radius $0$ )

Answer 1

Just take $(x,0,0,\ldots,0)$ where $x\in\Bbb R\setminus\Bbb Q$. Each point is a ball of radius $0$, clearly disjoint from every other ball, and the complement is certainly path connected.