Properties of Closed Sets
1. Ifis a set andis the closure ofthen
Letwhereis closed for eachThe intersection of any collection of closed sets is closed, Cl(A) is a closed set. A is closed if and only if Cl(A)=A so Cl(Cl(A))=Cl(A).
Letandwherefor eachandfor eachSinceeachcontainshence
On the other handis closed andsubsethence(2)
From (1) and (2)
is closed hence