Proof That for all Continuous Functions the Image of a Closed Set is a Subset of the Closure of the Image

Theorem

Supposeis continuous then

Proof

Supposeis continuous. Since

The setis closed andis continuous sois also closed so

Henceand

Suppose that forand letbe any closed subset ofand let

Hence

Sincewe haveand the inverse image of any closed subset ofis a closed subset ofHenceis continuous.