Page 172 - DMTH503_TOPOLOGY

Topology

Notes

Notes A regular T -space is called a T -space.
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A normal T -space is called a T -space.
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Examples of Regular Space

1. Every discrete space is regular.
2. Every indiscrete space is regular.

Example 4: Give an example to prove that a regular space is not necessarily a T -space.
1

Solution: Let X = {a, b, c} and let T = {T , X, {c}, {a, b}} be a topology on X.
The closed subsets of X are , X, {c}, {a, b). Clearly this space (X, T) satisfies the R-axiom and it is
a regular space. But it is not a T -space, for the singleton subset {b} is not a closed set.
1
Thus, this space (X, T) is a regular but not a T -space.
1

Example 5: Give an example of T -space which is not a T -space.
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Solution: Consider a topology T on the set  of all real numbers such that the T-nhd. of every
non-zero real number is the same as its -nhd but T-nhd. of 0 are of the form
{ 1 }

G - : n N
n
where G is a -nhd. of 0.

Then T is finer than .
Now, (R, ) is Hausdorff and  T, so (R, T) is Hausdorff.

{ 1 }
But : n N being T-closed, cannot be separated from 0 by disjoint open sets.

n
Hence, (R, T) is not a regular space.
Thus, (R, T) is T but not T .
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Theorem 1: A topological space (X, T) is a regular space iff each nhd. of an element x  X contains
the closure of another nhd. of x.
Proof: Let (X, T) be a regular space.

Then for a given closed set F and x  X such that x  F there exist disjoint open sets G, H such that
x  G and F  H.
Now x  G  G is a nhd. of x ( G is open)
Again, G  H = 

 G  X – H

 G  (X - H) = X – H (Since H is open and so X–H is closed)

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