Page 12 - DMTH503_TOPOLOGY

Topology

Notes (iii) G  T   {G : }  T
 1  1
G  T   {G : }  T
 2  2
Then (I) X  T T ,  T T by (i)
1 2 1 2
(II) A  T T , B T T  A  B  T  T
1 2 1 2 1 2
For A  T T , B T T
1 2 1 2
 A T , A T and B T , B T
1 2 1 2
 A B T , A B T by (ii)
1 2
 A B T T
1 2
(iv) G  T  T    
 1 2
 {G :  }  T  T
 1 2
For G  T  T  
 1 2
 G  T   and G  T  
 1  2
  G  T and  G  T by (iii)
 1  2
Thus, T  T is topology on X.
1 2
Part II: Let X = {a, b, c}. Then T = {X,  {a}} and T = {X, , {b}} are topologies on X.
1 2
Let G = {a}  T , G = {b}  T .
1 1 2 2
Then G  G = {a, b}  T  T .
1 2 1 2
Consequently T  T is not a topology on X.
1 2
Self Assessment

3. Prove that the intersection of an arbitrary collection of topologies for a set X is a topology
for X.

4. Let T be a topology on a set X  n ,  being an index set. Then  {T : r } is a topology
n n
on X.
1.3 Open Set, Closed Set and Closure of a Set

1.3.1 Definition of Open Set and Closed Set

Let (X, T) be a topological space. Any set A  T is called an open set and X-A is a closed set.

Example 8: If T = {, {a}, X} be a topology on X = {a, b} then , X and {a} are T-open sets.

Example 9: Let X = {a, b, c} and T = {, {a}, {b, c}, X} be a topology on X.
Since X – {a} = {b, c}

X – {b, c} = {a}
Therefore, T-closed sets are , {b, c} and X, which are the complements of T-open sets X, {b, c}, {a}
and  respectively.

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