Page 24 - DMTH503_TOPOLOGY

Topology

Notes Theorem 12: In any topological space, prove that
A = A  D(A)
Proof: Let (X, T) be a topological space and A  X.
To show: A = A  D(A)

Since A  D(A) is closed and hence
A  D(A) = A  D(A) …(1)
A  A  D(A)
 A  A  D(A) = A  D(A) [Using (1)]

A  A  D(A) …(2)
Now, We are to prove that
A  D(A)  A …(3)
But, A  A …(4)

To prove (3), we are to prove
D(A)  A …(5)
i.e., to show that
D(A)   {F  X : F is closed F  A} …(6)
i i i i
Let x  D(A) be arbitrary.
x  D(A)  x is a limit point of A
 x is a limit point of all those sets which contain A.
 x is a limit point of all those F appearing on R.H.S. of (6).
i
 x  D(F )  F ( F is closed)
i i i
 x  F for each i
i
 x  {F  X : F is closed}
i i i
 x  A

Thus any x  D(A)  x  A
D(A)  A
Hence the result (5) proved.
From (4) & (5), we get

A  D(A)  A  A = A
i.e., A  D(A)  A
Hence the result (3) proved.
Combining (2) & (3), we get the required result.

Self Assessment

12. Let X = {a, b, c} and let T = {, X, {b}, {c}}, find the set of all cluster points of set {a, b}.

18 LOVELY PROFESSIONAL UNIVERSITY