Page 72 - DMTH503_TOPOLOGY

Topology

Notes  x  G  A .
c
c
c
 A is the nhd of each of its point and therefore A is open.
Hence A is closed.

Example 9: Let (X, T) be a topological space and A  X. A point x of A is an interior point
of A iff it is not a limit point of X  A.

Solution: Let (X, T) be a topological space and A  X. Suppose a point x of A is an interior point of
A so that x  A, x  A°.
To prove that x is not a limit point of X  A i.e., x  D(X  A)

x  A  G  T with x  G s.t. G  A
 G  (X  A) = 



x
 G     (X A)   [ x (X A) ]

 G is an open set containing set.
G     (X A)  
x

This immediately shows that x  D(X  A).
Conversely suppose that (X, T) is topological space and A  X s.t. a point x of A is not a limit
point of (X  A).

To prove that x  A°.
By hypothesis x  A, x  D(X  A)

x  D(X  A)  G  T with x  G s.t. G     (X A)  
x

 G  (X  A) =  [ x  X  A]
 G  A.
 x  A  G  T with x  G s.t. G  A. This proves that x  A°.

Self Assessment

3. Let x be a topological space and let A, B be subset of x. Then.
(a)  =  or D() = 

(b) A  B  A  B or A  B  D(A)  D(B);
(c) x A   G {x} ;  
x
 
6.3 Summary

 A subset A of a topological space X is said to be closed if the set X - A is open.
 Let (X, T) be a topological space and A  X. A point x  X is said to be the limit point of A
if each open set containing x contains at least one point of A different from x.

 The set of all limit points of A is called the derived set of A and is denoted by D(A).

66 LOVELY PROFESSIONAL UNIVERSITY