Page 171 - DMTH503_TOPOLOGY

Unit 18: Normal Spaces, Regular Spaces and Completely Regular Spaces

bound of the set of distances d(x, a) from x to points a  A. Note that d(x, A)  0, and d(x, A) = 0 Notes
iff x is in the closure of A since d(X, A) = 0 is equivalent to saying that every ball B (x) contains
r
points of A.

Example 2: A compact Hausdorff space is normal.

Solution: Let A and B be disjoint closed sets in a compact Hausdorff space X. In particular, this
implies that A and B are compact since they are closed subsets of a compact space. By the
argument in the proof of the preceding example we know that for each x  A,  disjoint open sets
and V with x  and B  V . Letting x very over A, we have an open cover of A by the sets .
x x x x x
So, there is a finite subcover. Let be the union of the sets in this finite subcover and let V be
x
the intersection of the corresponding sets V . Then and V are disjoint open nhds. of A and B.
x
Example 3: A closed sub-space of a normal space is a normal space.
Solution: Let (X, T) be a topological space which is normal and (Y, ) a closed sub-space of (X, T)
so that Y is closed in X. To prove that Y is a normal space.
Let F , F  Y be disjoint sets which are closed in Y. Y is closed in X, a subset F of Y is closed in Y
1 2
iff F is closed in X.
 F and F are disjoint closed sets in X.
1 2
By the property of normal space (X, T).

 G , G  T s.t. F  G , F  G , G G = 
1 2 1 1 2 2 1 2
F  G  F  Y  G  Y  F = F  Y  G  Y
1 1 1 1 1 1 1
 F  G  Y.
1 1
Similarly F  G  F  G  Y.
2 2 2 2
By definition of relative topology,
G , G  T  Y  G , Y  G 
1 2 1 2
Also (G  Y)  (G  Y) = (Y  Y)  (G  G ) = Y   = .
1 2 1 2
Finally given a pair of disjoint closed sets F , F in Y,  disjoint sets.
1 2
G  Y, G  Y  s.t. F  G  Y, F  G  Y.
1 2 1 1 2 2
This proves that (Y, ) is a normal space.

Self Assessment

1. Show that if X is normal, every pair of disjoint closed sets have neighborhoods whose
closures are disjoint.
2. Give an example of a normal space with a subspace that is not normal.

3. Show that paracompact space (X, T) is normal.

18.2 Regular Space

A topological space (X, T) is said to be regular space if: given an element x  X and closed set
F  X s.t. x  F,  disjoint open sets G , G  X s.t. x  G , F  G .
1 2 1 2

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