Can we write every uncountable set $U$ as $V∪W$, where $V$ and $W$ are disjoint uncountable subsets of $U$?

Question

Is it true that for every uncountable set $U$, we can write $U=V∪W$, where $V$ and $W$ are disjoint uncountable subsets of $U$ ?

Answer 1

Yes, assuming the axiom of choice. Without the axiom of choice one might have an amorphous set. Such a set is uncountable in the sense that it is not finite, and there is no bijection between it and $\Bbb N$. However, it is not the disjoint union of any two infinite subsets.