Letbe the union of countably many closed sets:
where each
is closed.
The complement ofis
Applying De Morgans' rules, we obtain
Since eachis closed, the complement
is open.
is then an intersection of countably many open sets.
Letbe the intersection of countably many open sets:
where each
is open.
The complement of G is
Applying De Morgans' rules we obtains
Since eachis open,
is closed and
is the union of countably many many closed sets.