Page 26 - DMTH503_TOPOLOGY

Topology

Notes Exterior Operator: Let X be a topological space. Then, an exterior operator on X is a mapping
e : P(X)  P(X) satisfying the following postulates:
(i) e() = X, e(X) = 

(ii) e(A)  A
(iii) e[{e(A)}] = e(A)
(iv) e(A  B) = e(A)  e(B) where A and B are subsets of X.
Theorem 13: Prove that int(A) =  {G : G is open, G  A}.

or
Let X be a topological space and let A  X. Then, A° is the union of all open subsets of A.
Proof: Let x  A°  x is an interior point of A.
 A is a nhd of x.

Then  an open set G such that x  G  A and hence x   {G : G is an open subset of A}
Now let x   {G : G is open, G  A} …(1)
 x  some T-open set G which is contained in A
 x  A° by definition of A°

  {G : G is open, G  A}  A° …(2)
Thus from (1) and (2), we get
A° =  {G : G is open, G  A}
Theorem 14: Let X be a topological space and let A be a subset of X. Then int (A) is an open set.

Proof: Let x be any arbitrary point of int (A). Then x is an interior point of A.
This implies that A is a nhd. of x i.e.,  an open G such that x  G  A.
Since G contains a nhd of each of its points, it follows that A is a nhd of each of the point of G.
Thus, each point of G is a interior point of A.

Therefore, x  G  int (A).
Thus, it is shown that to each x  A°, these exists an open set G such that x  G  int (a).
Hence A° is a nhd of each of its point and consequently int (A) is open.
Theorem 15: Let X be a topological space and let A  X. Then A° is the largest open set contained
in A.
Proof: Let G be any open subset of A and let x be an arbitrary element of G i.e. x  G  A.
Thus A is a nhd of x i.e., x is an interior point of A.

Hence x  A°
 x  G  x  A°.
Thus G  A°  A.
Hence A° contains every open subset of A and it is, therefore, the largest open subset of A.

Theorem 16: Let X be a topological space and let A  X. Then A is open iff A° = A.
Proof: Let A be a T-open set.

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