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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