 Solution

Every point belongs to its component, so the components cover $X$; by part (ii) they are pairwise disjoint, and by part (iii) they are nonempty open intervals. Each contains a rational number. Fix an enumeration of $\mathbb{Q}$ and choose the first rational belonging to each component. Distinct components receive distinct rationals, giving an injection from the component family into a countable set. ThereforeThe empty open set is represented by the empty union.