Page 27 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

Since every T-open set is a T-nhd of each of its point, therefore every point of A is a T-interior Notes
point of A. Consequently A  A°,
Again, since each T-interior point of A belongs to A therefore A°  A.

Hence, A = A°
Consequently, if A = A°, then A must be a T-open set for A° is a T-open set.

1.7.3 Properties of Interior

Theorem 17: Let (X, T) be a topological space and A, B  X. Then
(i) ° = 
(ii) X° = X

(iii) A  B  A°  B°
(iv) (A°)° = A° or A°° = A°.
Proof: Let (X, T) be a topological space and A, B  X.

(i) & (ii), By definition of T, , X  T, consequently.
° = , X° = X
For A is open  A° = A.
(iii) Suppose A  B
any x  A°  x is an interior point of A.

  open set G s.t. x  G  A
 x  G  A  B x  G  B & G is open.
 x  B°

 A°  B°.
(iv) We Know that A° is open
Also G is open  G° = G …(1)
In view of this, we get
(A°)° = A° or A°° = A° (on putting G = A° in (1))

Theorem 18: Let i be an interior operator defined on a set X. Then these exists a unique topology
T on X s.t. for each A  X.
i(A) = T-interior of A.

Proof: Let i be an interior operator on X. Then a map
i : P(X)  P(X) s.t.
(i) i(X) = X

(ii) i(A)  A
(iii) i(A  B) = i(A)  i(B)
(iv) i[i(A)] = i(A), where A, B  X
P(X) being power set of X.

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