Proof That a Function is Continuous if and Only if the Inverse Image of a Closed Set is Closed

Theorem

A functionis continuous if and only if the inverse image of every closed subset ofis closed in

Proof

Ifis continuous then

Supposeis continuous andis any closed subset ofthenis an open subset ofandis an open subset of

Suppose for everyis closed inandis closed in

Letbe an open subset ofthenis closed inandis closed in

Henceis an open subset ofandis continuous.