Assume the countable sets are indexed by : given countable sets , list each as

Define by

taking the least index when an element lies in several so that is well-defined. As with , uniqueness of prime factorisation shows is an injection, so is countable.

is countable:

is a countable union of countable sets.

Any family of pairwise disjoint open intervals in is countable: those of length at least inject into the integer multiples of , so is a countable union of countable sets. Alternatively, each interval contains a rational, and picking one gives an injection .