Theorem
1. If
is a set and
is the closure ofthen
2.
3.
4.
Proof
Let
where
is closed for each
The 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).
Letand
where
for each
and
for each
Since
each
containshence
is closed.
and
so
(1)
On the other hand
is closed and
subsethence
(2)
From (1) and (2)
is closed hence