Can $\mathbb{R}$ be partitioned into $n$ dense sets with same cardinality? Are there sets $S_i\subseteq\mathbb{R}$ with $i\leq n$ such that


*

*$S_i$ are disjoint,

*$S_i$ have same cardinality,

*$S_i$ are dense in $\mathbb{R}$?


Write every number in base $n$. Then $S_1$ is the set of the numbers that have only a finite number of digits $\geq 2$. $S_2$ is the set of the numbers that are not in $S_1$ and have only a finite number of digits $\geq 3$, etc. $S_n$ is the set of the numbers that are not in any of the former.
