Page 163 - DMTH503_TOPOLOGY

Unit 17: The Separation Axioms

and x  H but x  H Notes
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Hence (X*, T*) is a T -space.
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Self Assessment

1. Show that any finite T -space is a discrete space. Is a discrete space T space? Justify your
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answer.
2. If (X, T) is a T -space and T is finer than T, then (X, T ) is also T -space.
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3. A finite subset of a T -space has no cluster point.
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4. If (X, T) is a T -space and T*  T, then (X, T*) is also a T -space.
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17.2 T -Axiom of Separation or Hausdorff Space
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A topological space (X, T) is said to satisfy the T -axiom or separation if given a pair of distinct
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points x, y  X.
 G, H  T s.t. x  G, y  H, G  x = 

In this case the space (X, T) is called a T -space or Hausdorff space or separated space.
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Example 5: Let X = {1, 2, 3} be a non-empty set with topology T = P(X) (all the subsets of
X, powers set or discrete topology). Hence
For 1, 2 1  {1}, 2  {1}
For 2, 3 2  {2}, 3  {2}

For 3, 1 3  {3}, 1  {3} and (X, T) is a T -space
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For 1, 2 1  {1}, 2  {2}  {1}  {2} = 
For 2, 3 2  {2}, 3  {3}  {2}  {3} = 
For 3, 1 3  {3}, 1  {1}  {3}  {1} = 

Example 6: Show that every T -space is a T -space.
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Solution: Let (X, T) be a T -space.
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Let x, y be any two distinct points of X. Since the space is T , then there exist open nhd. G and H
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of x and y respectively such that G H = .
Thus G and H are open sets such that
x  G but y  G
and y  H but x  H
Hence the space is T .
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Example 7: Prove that every T -space is a T -space but converse is not true. Justify.
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Solution: Let (X, T) be a T -space.
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Let x, y be any two distinct points of X.

Since the space is T ,  open nhds G and H of x and y respectively such that G H = 
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