Page 11 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

(ii) Let G , G  T Notes
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 G , G are countable
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 G ,  G is countable
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 (G  G ) is countable (by De-Morgan’s law)
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 G  G  T
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(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 .
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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}}
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T  T = {, X, {1}} is a topology on X.
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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
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a topology on X but T  T is not a topology on X.
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Solution: Part I: Let T , T be two topologies on the same set X.
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We are to prove that T  T is a topology on X.
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By assumption,

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

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