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Introduction
The Urysohn Metrization Theorem
Appendix
Lecture : The Urysohn Metrization Theorem
Dr. Sanjay Mishra
Department of Mathematics
Lovely Professional University
Punjab, India
November 22, 2014
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
Outline
1
Introduction
2
The Urysohn Metrization Theorem
3
Appendix
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
Introduction I
The Urysohn Metrization Theorem is
“ If a topological space X is regular and has a countable basis, then X
is metrizable”.
This theorem give us conditions under which a topological space is
metrizable. The proof of this theorem concern following concepts
as like metric topologies, countability and separation axioms.
There are two versions od the proof and both are useful
generalization. The first version generalized to give an imbedding
theorem for completely regular spaces.
The second version will be generalized when we prove
Nagata-Smirnove Metrization theorem.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
Introduction II
The idea behind the proof, however, is straightforward. Using the assumptions that X is regular and has a countable basis, it can be shown
that X can be embedded in a metric space. Therefore X is homeomorphic to a subspace of a metric space. Since a subspace of a metric space
is metrizable, and since metrizability is a topological property, it follows
that X is metrizable.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem I
Theorem
Every regular space with countable basis is metrizable space.
Proof:
We shall prove that X is metrizable by imbedding X is a metrizable
space Y i.e. by showing X homeomorphic with a subspace of Y .
The two version of the proof differ in the choice of the metrizable space
space Y .
In the first version, Y is the space Rω in the product topology and as
know that this space is metrizable.
In the second version, the space Y is also Rω , but this time in the
topology given by the uniform metric ρ̄.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem II
In each case, it turns out that our construction actually imbeds X in the
subspace [0, 1]ω of Rω .
Step I: We will prove that: There exists a countable collection of continuous function fn : X → [0, 1] having the property that given any point
x0 of X and any neighborhood U of x0 , there exists an index n such that
fn is positive at x0 and vanishes outside U i.e.
> 0, x0 ∈ U ;
fn (x0 )
= 0, x0 ∈
/ U.
It is a consequence of Urysohn’s lemma that given x0 and U , there exists
such a function. However, if we choose one such function for each pair
(x0 , U ), the resulting collection will not in general be countable. Our
task is to cut the collection down to size. Here is one way to proceed:
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem III
Let {Bn } be a countable basis for X. For each pair of n, m indices
for which B¯n ⊂ Bm , apply the Urysohn lemma to choose a continuous
function gn,m : X → [0, 1] such that gn,m (B¯n ) = {1} and gn,m (X −Bm ) =
{0}. Then the collection gn,m satisfies our requirement:
Given x0 and given open set U of x0 , one can choose basis element Bm
containing x0 that is contained in U .
Using regularity, one can then choose Bn so that x0 ∈ Bn and B¯n ⊂ Bm .
Then n, m is a pair of indices for which the function gn,m is defined; and
it is positive at x0 and vanishes outside U .
Because the collection {gn,m } is indexed with subset of Z+ × Z+ , it is
countable; therefore it can be reindexed with the positive integers, giving
us the desired collection {fn }.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem IV
Step II:(First version of the proof): Given the function fn of Step - I,
take Rω in the product topology and define a map F : X → Rω by the
rule
F (x) = (f1 (x), f2 (x), . . .)
Now we want to show that F is an imbedding.
First, F is continuous because Rω as the product topology and each fn
is continuous.
Second, F is injective because given x 6= y, we know there is an index n
such that fn (x) > 0 and fn (y) = 0; therefore F (x) 6= F (y).
Finally, we will show that F is a homomorphism of X onto its image,
the subspace Z = F (X) of Rω .
We know that F define a continuous bijection of X with Z, so we need
only show that for each open U in X, the set F (U ) is open in Z.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem V
Let z0 be a point of F (U ). We shall find an open set W of Z such that
z0 ∈ W ⊂ F (U )
Let x0 be the point of U such that F (x0 ) = z0 . Choose an index N for
which fN (x0 ) > 0 and fN (X − U ) = {0}. Take the open ray (0, +∞) in
R, and let V be the open set
−1
V = πN
((0, +∞))
of Rω .
Let W = V ∩Z; then W is open in Z (by definition of subspace topology).
Now we want to show that z0 ∈ W ⊂ F (U ).
First, z0 ∈ W and this is because
πN (z0 ) = πN (F (x0 )) = fN (x0 ) > 0
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem VI
Second, W ⊂ F (U ).
For if z ∈ W , then z = F (x) for some x ∈ X, and πN (z) ∈ (0, +∞).
Since πN (z) = πN (F (x)) = fN (x), and fN vanishes outside U , the point
x must be in U . Then z = F (x) is in F (U ), as desired. Thus F is an
imbedding of X in Rω .
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
The Urysohn Metrization Theorem VII
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
Let A be a non-empty collection of sets.
Indexing Function: An indexing function for A is a surjective
function f from some set J, called index set, to A. The collection
A, together with the indexing function f , is called an indexed
family of sets. We shall denote the indexed family itself by the
symbol {Aα }α∈J which is read ”the family of all Aα , as α ranges
over J.”
Note: An indexing function is required to be surjective, it is not
required to be injective.
The types of useful index set are the set {1, . . . , m} of positive
integers from 1 to m. And the set Z+ of all positive integers.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
m-tuple: Let m be a positive integer. Given a set X, we define
m-tuple of elements of X to be function
x : {1, . . . , m} → X
If x is an m-tuple, we denote the value of x at i by the symbol xi
rather that x(i) and call it the ith coordinate of x. And we often
denote the function x itself by the symbol
(x1 , . . . , xm ) or (xm )m∈J , where J = {1, . . . , m}
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
ω-tuple: Given a set X, we define ω-tuple of elements of X to be
function
x : Z+ → X
We call such function a sequence, or infinite sequence, of elements
of X. If x is an ω-tuple, we often denote the value of x at i by
the symbol xi rather that x(i) and call it the ith coordinate of x.
We denote the function x itself by the symbol
(x1 , x2 , . . .) or (xm )m∈Z+
Rm and Rω : Rm denotes the set of all m-tuple of the real
numbers and it is called Euclidean m-space. Similarly, Rω is set
of all ω-tuple (x1 , x2 , . . .) of real numbers called
infinite-dimensional Euclidean space.
Sanjay Mishra
The Urysohn Metrization Theorem
Introduction
The Urysohn Metrization Theorem
Appendix
An embedding (or imbedding) is one instance of some mathematical
structure contained within another instance, such as a group that is a
subgroup.
When some object X is said to be embedded in another object Y ,
the embedding is given by some injective and structure-preserving map
f : X → Y . The precise meaning of “structure-preserving” depends on
the kind of mathematical structure of which X and Y are instances.
Theorem (embedding (or imbedding))
Let A, B be topological spaces. Let f : A → B be a mapping and the
image of f be given the subspace topology.
Let the restriction f |A×f (A) of f to its image be a homeomorphism.
Then f is an embedding (of A into B).
Sanjay Mishra
The Urysohn Metrization Theorem