Can $\mathbb{R}$ be written as an uncountable union of disjoint uncountable subsets?
I was thinking of the following: Consider an uncountable proper subfield $F$ of $\mathbb{R}$, then consider $\mathbb{R}$ as a vector space over $F$. If we accept the axiom of choice, this vector space has a basis, say, $B=\left\{b_\alpha:\alpha\in I\right\}$.
However, I am not sure if $B$ itself can be uncountable. If it can, we may consider the sets $$ S_\alpha=\left\{\operatorname{span}_F(b_\alpha)\setminus\left\{0\right\}: \alpha\in I\right\} $$ Then $\bigcup\limits_{\alpha\in I}S_\alpha \subset \mathbb{R}$, and we can easily obtain the desired form.
