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.
Also
so that
is the intersection of countably many open sets.
The set of rational numbers
is countable and each rational number
is a singleton set
in 
Each singleton set is closed in
sois the union of closed sets.is also countable so
is the union of a countable number of closed sets.
