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 $\mathbb{R}$?


Split $\mathbb{Q}$ into $n$ dense subsets, where $p/q$ in lowest terms goes into the subset $p\pmod n$.  Let the $i$th set be the union of the irrational numbers $\{x\in\mathbb{R}\backslash\mathbb{Q}|\lfloor x\rfloor=i\pmod n\}$ with one of the rational subsets.
