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 $\mathbb{R}$?


Partition the set of cosets of $\mathbb{Q}$ into $n$ sets of equal uncountable cardinality, and take the union of each partition element. The resulting partition of $\mathbb{R}$ consists of $n$ sets, each contains a coset of $\mathbb{Q}$ and so is dense, and their cardinalities are equal.
