Download Completely Regular Space

Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Lecture : Completely Regular Space
Dr. Sanjay Mishra
Department of Mathematics
Lovely Professional University
Punjab, India
November 16, 2014
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Outline
1
Introduction
2
Separation of subset of Space by Continuous map
3
Completely Regular Space
4
Tychonoff Space
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Introduction I
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Definition
If A and B are two subsets of the topological space X, and if there is a
continuous function f : X → [0, 1] such that f (A) = {0} and
f (B) = {1}, we say that A and B can be separated by a continuous
functions.
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Completely Regular Space I
Definition
A space X is said to completely regular if given closed subset C of X
and a point x ∈ X such that x ∈
/ C, then there exists a continuous map
f : X → [0, 1] such that f (x) = 0 and f (C) = {1}
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Completely Regular Space II
1
A normal space is completely regular by Urysohn lemma.
2
A completely regular space is regular.
Since given f , the sets f −1 [0, 21 ) and f −1 ( 12 , 1] are disjoint open set
about A and x0 , respectively.
3
This new axiom is fit in between regularity and normality.
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Completely Regular Space III
Unlike normality, the new separation axiom is nicely behaved with regard
to subspace and products.
Theorem
A subspace of a completely regular space is completely regular. A
product of completely regular space is completely regular.
Proof:
Let (X, τ ) be completely regular space and (Y, τ1 ) be a subspace of X.
Now we want to show that Y is also completely regular space.
Let y ∈ Y and C be a τ1 -closed subset of Y such that y ∈
/ C. Because
Y ⊂ X, so y ∈ X.
By definition of subspace topology, if C is τ1 -closed show that there
exists τ -closed set C1 such that C = C1 ∩ Y .
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Completely Regular Space IV
And y ∈
/C⇒y∈
/ C1 ∩ Y ⇒ y ∈
/ C1 .
Now y ∈ X and C1 is τ -closed set such that y ∈
/ C1 , then by the property
of completely reguar space, there exist a continuous map f : X → [0, 1]
such that
f (y) = 0, f (C1 ) = {1}
Let g denote restriction of f to Y , then g : Y → [0, 1] is a continuous
map such that
g(u) = f (u) ∀ u ∈ Y
⇒
g(y) = f (y) = 0, y ∈ Y ⇒ g(y) = 0
Sanjay Mishra
Completely Regular Space
(1)
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Completely Regular Space V
g(u) = f (u) ∀ u ∈ C ⊂ Y
Again
f (u) = 1 ∀ u ∈ C1 and C ⊂ C1
But
f (u) = 1 ∀ u ∈ C
∴
By the result (1) and (2)
g(u) = 1
∀ u ∈ C or g(C) = {1}
Finally, g : Y → [0, 1] is a continuous map such that
g(y) = 0, g(C) = {1}
This proves that (Y, τ1 ) is completely regular space.
Sanjay Mishra
Completely Regular Space
(2)
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Tychonoff Space I
Definition (Tychonoff Space or T
1 -space)
32
A completely regular T1 -space is said to be a Tychonoff space.
Remark
It may be noted that since product of T1 -space is T1 -space and product
of completely regular space is completely regular space, so product of
Tychonoff space is Tychonoff space.
Sanjay Mishra
Completely Regular Space
Introduction
Separation of subset of Space by Continuous map
Completely Regular Space
Tychonoff Space
Tychonoff Space II
Some Important Results
1
Every completely regular space is a regular space as well.
2
Every completely regular T1 -space is Hausdorff space or T2 -space.
3
Every subspace of a completely regular space is completely regular
space.
4
Product of completely regular space is a completely regular space.
5
Every subspace of Tychonoff space is Tychonoff space.
6
Every Tychonoff space is Hausdorff space.
Sanjay Mishra
Completely Regular Space