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