Page 173 - DMTH503_TOPOLOGY

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

Notes
 G  X – H
 G  X – F

 G  X – F = M (say)

 G  M.
Since F is a closed set, M is an open set and
x  F  x  X – F.
 x  M, thus M is a nhd. of x.

Hence, if M is a nhd. of x, there exists a nhd. G of x such that
x  G  G  M.

Conversely, Let N and N be the nhds. of x  X.
1 2
If N  N , then we have to show that (X, T) is a regular space.
2 1
Let F be a closed subset of X and let x be an element of X such that x  F.
Now F is closed and x  F.
 x  X – F and X – F is open.
 X – F is a nhd. of x.

Let X – F = N , then by hypothesis
1
x  N  N  X – F ( N  N )
2 2 2 1
Let us write N = G and
2 1
X – N = G
2 2

Then G  G = N  (X – N )
1 2 2 2
= (N  X) – (N  N )
2 2 2
= N – N
2 2
= .

 G  G = .
1 2
Also x  N  x  G
2 1
and N  X – F  F  X – N
2 2
or F  G
2
Since N is a closed set, therefore G is open.
2 2
Thus, we have proved that for a given closed subset F of X and x  X such that x  F there exist
disjoint open subsets G , G such that
1 2
x  G, and F  G
2
Hence X is a regular space.

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