Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / E / E3 / b / iv / Solution

The components cover $X$ and are pairwise disjoint by (b)(ii); by (iii) they are open intervals. Each nonempty interval contains a rational number. Enumerate the rationals and assign to each interval the first rational it contains. Disjointness makes these assigned rationals distinct, giving an injection from the set of components into $\mathbb{N}$. Thus $\boxed{X\text{ is a countable union of pairwise disjoint open intervals}}$, with the empty union for $X=\varnothing$.