## Proof That for a Closed Function the Closure of an Image Set is a Subset of the Image of the Closure and Vice Versa

Theorem

The statementsis a closed function andare equivalent.

Proof

Supposeis a closed function. Letrepresent any subset ofthenis a closed set andis closed.

Sincefor allwe have

Suppose now that for each set

Ifis a closed set thenand

Thereforesois closed andis a closed function.