= 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 $X=\varnothing$.
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