Theorem

1. Ifis a set andis the closure ofthen

2.

3.

4.

Proof

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

is closed.andso(1)

On the other handis closed andsubsethence(2)

From (1) and (2)

is closed hence