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Unit 1: Topological Spaces




          (ii)  Let G , G   T                                                                  Notes
                   1  2
                   G , G  are countable
                      1  2
                   G ,  G  is countable
                      1    2
                   (G   G ) is countable                         (by De-Morgan’s law)
                      1   2
                   G  G   T
                     1   2
          (iii)  Let {G  :   } be an arbitrary collection of members of sets in T.
                    
                   G  is countable     
                      
                   G  :  } is countable
                        
                   G  :  }’ is countable                   (by De-Morgan’s law)
                        
                   G  :  }  T
                        
               Hence, T is a topology for X.

          Self Assessment

          1.   Construct three topologies T , T , T  on a set X = {a, b, c} s.t. T   T  T .
                                      1  2  3                   1   2   3
          2.   Let X = {a, b, c} and T = {, X, {b}, {a, b}. Is T is a topology for X?

          1.2 Intersection and Union of Topologies


          Intersection of any two topologies on a non-empty set is always topology on that set. While the
          union of two topologies may not be a topology on that set.


                 Example 6: Let X = {1, 2, 3, 4}
             T  = {, X, {1}, {2}, {1, 2}}
              1
             T  = {, X, {1}, {3}, {1, 3}}
              2
             T   T  = {, X, {1}} is a topology on X.
              1   2
             T   T  = {, X, {1}, {2}, {3}, {1, 2}, {1, 3}} is not a topology on X.
              1   2

                 Example 7: If T  and T  are two topologies defined on the same set X, then T   T  is also
                             1    2                                          1   2
          a topology on X but T   T  is not a topology on X.
                            1   2
          Solution: Part I: Let T , T  be two topologies on the same set X.
                           1  2
          We are to prove that T   T  is a topology on X.
                            1   2
          By assumption,

          (i)  X  T , X  T
                   1      2
                 T ,   T
                   1     2
          (ii)  A, B T   A B T
                      1          1
               A, B T   A B T
                      2          2





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