TITLE: Can $\mathbb R$ be written as the disjoint union of (uncountably many) closed intervals?
QUESTION [5 upvotes]: From this post: http://terrytao.wordpress.com/2010/10/04/covering-a-non-closed-interval-by-disjoint-closed-intervals/
We know that $\mathbb R$ can't be written as the countable union of disjoint closed intervals. Can we do it if we allow uncountably many intervals? There doesn't seem to be a nice way to construct such a cover, since if we choose a closed interval, we split the line into two remaining pieces to cover that re still homeomorphic to the original real line. But on the other hand, I feel as if it should be possible, since with the possibility of uncountably many intervals, you should be able to find a way to cover everything.

REPLY [13 votes]: No.
Since $\mathbb{Q}$ is dense in $\mathbb{R}$, within each interval $[a_i, b_i]$, we can select a single $q_i \in \mathbb{Q}$ with which
to "label" the interval.  The set of $q_i$'s is a subset of $\mathbb{Q}$, which is countable.