Page 14 - DMTH503_TOPOLOGY

Topology

Notes
(iv)  A   B  A  B

(v)  A B   A  B

(vi) A  A

Proof:
(i) Since  and X are open as well as closed.
So, , X being closed, we have

 = , X = X

(ii) Since we know that A is the smallest T-closed set containing A so A  A
(iii) Let A  B

Then A B  B

i.e. B is a closed superset of A. ( B  B )
But A is the smallest closed superset of A.

 A  B

Thus, A B  A  B .

(iv) We have A  A  B  A  A  B by (iii)
and B  A  B  B  A  B by (iii)

Hence A  B   A   B … (I)

Since A , B are closed sets, A  B is also closed.

 A  B  A  B … (II)
From (1) & (2), we have A  B  A  B .

(v) We have

(A  B) A  A  B  A by (iii)

and (A  B) B  A  B  B by (iii)

Hence A  B  A  B .

(vi) We know that if A is a T-closed subset then A = A by the theorem: In a topological space
(X, T) if A is subset of X then A is closed iff A = A.

But A is also a T-closed subset.

 A = A.

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