Every point belongs to its component, so the components cover ; 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 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. Therefore
(1)
The empty open set is represented by the empty union.

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