Proof That the Complement of a Union of Countably Many Closed Sets is the Intersection of Countably Many Open Sets and Vice Versa

Letbe the union of countably many closed sets:where eachis closed.

The complement ofis

Applying De Morgans' rules, we obtain

Since eachis closed, the complementis open.is then an intersection of countably many open sets.

Letbe the intersection of countably many open sets: where eachis open.

The complement of G is

Applying De Morgans' rules we obtains

Since eachis open,is closed andis the union of countably many many closed sets.

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