Page 174 - DMTH503_TOPOLOGY

Topology

Notes Theorem 2: Prove that a normal space is a regular space i.e. to say, X is a T -space  X is a T -space.
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Proof: Let (X, T) be a T -space so that
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(i) X is a T -space
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(ii) X is a regular space

To prove that X is a T -space. For this we must show that
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(iii) X is a T -space
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(iv) X is a regular space
Evidently (i)  (iii)

If we show that (ii)  (iv), the result will follow. Let F  X be a closed set and x  X s.t. x  F.
X is a T -space  {x} is closed in X.
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By normality, given a pair of disjoint closed sets {x} and F in X,  disjoint open sets G, H in X s.t.
{x}  G, F  X, i.e. given a closed set F  X and x  X s.t. x  F.  disjoint open sets G, H in X s.t.
{x}  G, F  H. This proves that (X, T) is a regular space.

Example 6: Show that the property of a space being regular is hereditary property.
Solution: Let (Y, ) be a subspace of a regular space (X, T). We claim that the property of regularity
is hereditary property. If we show that (Y, ) is regular, the result will follow.
Let F be a -closed set and p  Y s.t. p  F.

È
T
Let F = closure of F w.r.t. the topology T. and F = closure of F w.r.t. the topology we know
È
T
that F = F  Y.
T
È
Since F is a -closed set  F = F  F = F  Y.
T
T
p  F  p  F  Y  p  F or p  Y
T
 p  F for p  Y.
T
F is a T-closed set.
closure of any set is always closed.
T
T
By the regularity of (X, T), given a closed set F and a point p  X. s.t. p  F ;  disjoint sets G,
T
H  T with p  G, F  H.
T
Consequently, F = F  Y  H  Y, p  G  Y
(G  Y)  (H  Y) = (G  H)  (Y  Y) =   Y = 
Thus, we have shown that given a -closed set F and a point p  Y s.t. p F, we are able to find
out the disjoint open sets G  Y, H  Y in Y s.t. p  G  Y, F  H  Y.
This proves that (Y, ) is regular. Hence proved.

Self Assessment

4. Show that the usual topological space (R, È) is regular.
5. Show that every T -space is a T -space.
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