Proof That for a Closed Function the Closure of an Image Set is a Subset of the Image of the Closure and Vice Versa
The statementsis a closed function andare equivalent.
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.