## Proof That a Closed Interval in R is the Union of Countably Many Closed Sets and the Intersection of Countably Many Open Sets

A closed interval is a closed set, hence it is the union of a single closed set, namely itself.

Alsoso thatis the intersection of countably many open sets.

The set of rational numbersis countable and each rational numberis a singleton setin Each singleton set is closed insois the union of closed sets.is also countable sois the union of a countable number of closed sets.