Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected.

Let $\frak P$ be a partition of $\mathbb{R}^n$ into *closed* sets and assume $\mathbb{R}^n\notin{\frak P}$ (that is, ${|\frak P|}>1$). Let ${\frak P}_0\subseteq {\frak P}$ consist of the elements of ${\frak P}$ that have Lebesgue measure $0$. Is it necessarily true that $|{\frak P}_0| = 2^{\aleph_0}$?