Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Introduction Countable Space Results on Countable Spaces Appendix Lecture : Countability Axioms for Topological Spaces Dr. Sanjay Mishra Department of Mathematics Lovely Professional University Punjab, India October 28, 2014 Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Outline 1 Introduction 2 Countable Space First Countable Space Second Countable Space Lindelof Space 3 Results on Countable Spaces 4 Appendix Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Introduction I The concept which we are going to discuss are unlike compactness and connectedness that do not arise naturally from the study of calculus nd analysis. This concept is for deeper study of topology itself. Such problems as imbedding a given space in a metric space or in compact Housdorff space are basically problems of topology rather than analysis. These particular problems have solutions that involve the countability axioms. Our basic goal is to prove the “Urysohn metrization theorem”. That is if a topological space X satisfy a certain countability axiom (second) and a certain separation axiom (regular), then the X can be imbedded in a metric space and is thus metrizable. So, countability axiom will play very important role in proving of this theorem. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space First Countable Space I Definition (First Countable Space) A topological space X is called first countable space is it satisfies the following axiom called the first axiom of countability. “ For each point p ∈ X there exists a countable base Bp of open sets containing p such that every open set G containing p also contains a member of Bp . Or in other words, a topological space X is first countable space if and only if there exists a countable local base at every point p ∈ X. Remark The first countable axiom is a local property of a topological space X, i.e. if depends only upon the properties of arbitrary neighborhoods of the point p ∈ X. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space First Countable Space II Example Let X be metric space and let p ∈ X. Recall that the countable class of open spheres {S(p, 1), S(p, 12 ), S(p, 13 ), . . .} with center p is a local base at p. Hence every metric space satisfies the first axiom of countability. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space First Countable Space III Example Let X be any discrete space. Now the singleton set {p} is open and is contained in every open set G containing p ∈ X. hence every discrete space satisfies first axiom of countability. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space First Countable Space IV Example Let X = N and J = {X, φ, {1}, {1, 2}, . . . , {1, 2, . . . , n}, . . .} then obviously (X, J ) is a first countable topological space. Note that this is not an interesting example of a first countable topological space. Once the topology J is a countable collection then (X, J ) is a first countable space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space First Countable Space V Example Let X = R and Jl be the lower limit topology on R generated by {[a, b) : a, b ∈ R, a < b}. For each x ∈ X, Bx = {[x, x + n1 ) : n ∈ N} is a countable local base at x. Hence (R, Jl ) = Rl is a first countable topological space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space Second Countable Space I Definition (Second Countable Space) A topological space (X, τ ) is called a second countable space if it satisfies the following axiom, called second axiom of countability. “There exists a countable base B for the topology τ ”. Remark Second countability is a global rather than a local property of a topological space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space Second Countable Space II Example The class B of open intervals (a, b) with rational endpoints, i.e a, b ∈ Q, is countable and is a base for the usual topology on the real line R. Thus R is a second countable space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space Second Countable Space III Example Consider the discrete topology D on R. Recall that a class B is base for a discrete topology if an only if if contains all singleton sets. But R, and hence the class of singleton subsets {p} of R, are non-countable. Accordingly, (R, D) does not satisfy the second axiom of countability. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space Second Countable Space IV Comparison between first and second countable spaces First countable space For each point p ∈ X there exists a countable base Bp of open sets containing p such that every open set G containing p also contains a member of Bp . Second countable space There exists a countable base B for the topology τ . It is global property of space. It is local property of space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space Lindelof Space I A Lindelof space is a topological space in which every open cover has a countable subcover. The Lindelof property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix First Countable Space Second Countable Space Lindelof Space Lindelof Space II Definition A topological space (X, τ ) is said to be aSLindelof space if for any collection A of open sets such that X = A∈A A there S exists a countable subcollection say B ⊂ A such that X = B∈B B. or we can say that (X, τ ) is called Lindelof space f every cover of X is reducible to a countable cover. From this definition it show that “ Every compact space is Lindelof space but its converse is not true”. For example a discrete topological space (X, τ ) is Lindelof space where X is any infinite countable set but it is not compact. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces I Now here we will discuss some important results on countable spaces as like, Relation between first and second countable spaces. Hereditary property of countable space. Subspace of countable space. Countable product of topological spaces. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces II Theorem A second countable space is always first countable space. Proof: Here we want to show that Second countable space ⇒ First countable space. As given that space (X, τ ) is second countable space, then ⇒∃ a countable base B for τ on X ⇒B ∼ N ⇒B can be expressed as B = {Bn ∈ B : x ∈ Bn } Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces III Let x ∈ X be arbitrary and a new collection of members of B is Lx = {Bn ∈ B : x ∈ Bn } Now for Lx we can say that 1 Since subset of countable set B is countable so the collection Lx will be countable. 2 Because Lx ⊂ B and members of B are τ -open sets, so members of Lx will be also τ -open sets. 3 Any G ∈ Lx ⇒ x ∈ G, according to the construction of Lx . Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces IV 4 Let G ∈ τ be arbitrary such that x ∈ G. Then, by definition of base x ∈ G ∈ τ ⇒Br ∈ B such that x ∈ Br ⊂ G ⇒∃ Br ∈ Lx such that Br ⊂ G For Br ∈ B with x ∈ Br ⇒ Br ∈ Lx . Finally, we can say that x ∈ G ∈ τ ⇒ ∃ Br ∈ Lx such that Br ⊂ G Hence we can say that Lx is a countable local base at x ∈ X. So, by definition X is first countable space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces V Theorem First countable space does not imply second countable space. Proof: Let τ be a discrete topology on infinite set X so that every subset of X is open in X and hence in particular, each singleton set {x} is open in X for each x ∈ X. Write B = {{x} : x ∈ X}. Here we can verify that B is a base for τ on X and B is uncountable. For X is uncountable. Hence X is not second countable space. If we take Lx = {x}, then evidently Lx is a countable local base at x ∈ X as it has only one member. For any G ∈ τ with x ∈ G, there exists {x} such that x ∈ {x} ⊂ G. From definition, X is first countable but not second countable space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces VI Here we will discuss the hereditary property of countable spaces. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces VII Theorem The property of a space being first countable is hereditary. Proof: Let (Y, τ1 ) be a subspace of first countable (X, τ ), then we want to prove that (Y, τ1 ) is also first countable space. Let y ∈ Y ⊂ X be arbitrary. If X is first countable space show that there exists a countable local base x ∈ X. Consider countable local base B = {Bn : n ∈ N} for y ∈ X. Then y ∈ Bn , ∀ n ∈ N. Write B1 = {Y ∩ Bn : n ∈ N} (1) Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces VIII y ∈ Y, y ∈ Bn ∀ n ∈ N ⇒y ∈ Y ∩ Bn , ∀ n ∈ N Bn ∈ B ∀ n ∈ N ⇒Bn ∈ τ ⇒ Y ∩ Bn ∈ τ1 Now we claim that B1 is a countable local base at y for τ1 on Y . 1 Since by equation (2) there is bijection map n → Y ∩ Bn exists between N and B1 , so N ∼ B1 . Hence B1 is countable. 2 For any G ∈ B1 ⇒ y ∈ G. 3 The collection B1 is family of all τ1 -open sets. Sanjay Mishra Countability Axioms (2) (3) Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces IX 4 Let G ∈ τ1 be arbitrary such that y ∈ G, then there exists H ∈ τ such that G = H ∩ Y , (by definition of subspace). As assume that y ∈ Y , y ∈ G and G = H ∩ Y ⇒ y ∈ H. By definition of local base for X y ∈ H ∈ τ ⇒∃ Br ∈ B such that y ∈ Br ⊂ H y ∈ H ∩ Y ∈ T1 ⇒∃ Br ∩ Y ∈ B1 such that Br ∩ Y ⊂ H ∩ Y y ∈ G ∈ τ1 ⇒Br ∩ Y ∈ B1 such that y ∈ Br ∩ Y ⊂ G (4) Finally we can say that B1 is a local base at y ∈ Y for topology τ1 on Y , hence (Y, τ1 ) is first countable space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces X Theorem The property of a space being second countable space is hereditary. Proof: Let (Y, τ1 ) be a subspace of topological space (X, τ ) which is second countable space. We want to proof that Y is also second countable space. By given, there exists a countable base B for τ on X. B is countable ⇒ B ∼ N ⇒ B is expressible as B = {Bn : n ∈ N} Write B1 = {Y ∩ Bn : n ∈ N} (5) Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces XI 1 Since by equation (5) there is bijection map n → Y ∩ Bn exists between N and B1 , so N ∼ B1 . Hence B1 is countable. 2 The collection B1 is family of all τ1 -open sets. For Bn ∈ B ⇒Bn ∈ τ ∵B⊂τ ⇒Y ∩ Bn ∈ τ1 ∵ Y is subspace of X. 3 Any y ∈ G ∈ τ1 ⇒ ∃ Br ∩ Y ∈ B1 such that y ∈ Y ∩ Br ⊂ G. For proving this let G ∈ τ1 such that y ∈ G, then there exists H ∈ τ such that G = H ∩ Y . y ∈ G ⇒ y ∈ H ∩ Y ⇒ y ∈ H, y ∈ Y By definition of base, any y ∈ H ∈ τ ⇒ ∃ Br ∈ B such that y ∈ Br ⊂ H Sanjay Mishra Countability Axioms (6) Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces XII From which any y ∈ H ∩ Y ∈ τ1 ⇒ ∃ Y ∩ Br ∈ B1 such that y ∈ Y ∩ Br ⊂ G. i.e. any y ∈ G ∈ τ1 ⇒ ∃ Y ∩ Br ∈ B1 such that y ∈ Y ∩ Br ⊂ G. From the definition and above results we can say that B1 is a countable base for τ1 on Y . Consequently (Y, τ1 ) is second countable space. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces XIII Theorem A second countable space is Lindelof space. Or Let (X, τ ) is second countable space. Let O an open cover of a set A ⊂ X. Then O is reducible to a countable cover. Proof: As given (X, τ ) is a second countable space and O be an open cover A ⊂ X, so that A ⊂ ∪{O : O ∈ O} (7) Here we want to show that O is reducible to a countable cover. As X is countable space so, there exists a countable base B for τ on X. Since B is countable and hence its members may be enumerated as B1 , B2 , . . .. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Results on Countable Spaces XIV Let x ∈ A be arbitrary, then equation (7) ⇒ x ∈ Ox for at least one Ox ∈ O. Since Ox ⊂ X is open, then Ox ∈ τ . By definition of base x ∈ Ox ∈ τ ⇒ ∃Bx ∈ B such that x ∈ Bx ⊂ Ox We can write A ⊂ ∪{Bx : x ∈ A} (8) (9) ∴ ∪{Bx : x ∈ A} ⊂ B and hence it can be expressed as A ⊂ ∪{Bn : n ∈ N} (10) Choose On ∈ O such that Bn ⊂ On ∀n ∈ N ⇒ ∪{Bn : n∈N } ⊂ ∪{On : n ∈ N} In this event (10) reduces to A ⊂ ∪{On : n ∈ N} This show that the open cover O of A reducible to a countable cover {On : n ∈ N}. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Definition (Finite and Infinite Set) A set X is finite if X is empty or if there is a bijection f : {1, 2, . . . , n} → X for some n ∈ Z+ . A set that is not finite is said to be infinite. A special type of infinite set is the countably infinite set, a prototype of which is the set of positive integers Z+ . Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Definition (Countable and Uncountable Set) A set X is countably infinite if there is a bijection f : Z+ → X. A set that is either finite or countably infinite is said to be countable. A set that is not countable is said to be uncountable. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Definition (Local Base) Let (X, τ ) be a topological space. A family Bx of open subsets of X is said to be a local base at x ∈ X for τ on X if 1 Any B ∈ Bx ⇒ x ∈ B. 2 Any G ∈ τ with y ∈ G ⇒ ∃ B ∈ Bx such that y ∈ B ⊂ G. Example Let x ∈ R be arbitrary. And An = (x − n1 , x + n1 ) ∀ n ∈ N Take Bx = {An : n ∈ N}. Evidentaly Bx is a local base at x ∈ R for usual topology on R. Sanjay Mishra Countability Axioms Introduction Countable Space Results on Countable Spaces Appendix Theorem 1 A subset of a finite set is a finite set. 2 A finite union of finite sets is a finite set. 3 A product of finite sets is a finite set. 4 A subset of a countable set is a countable set. 5 A countable union of countable sets is a countable set. 6 A product of countable sets is a countable set. Sanjay Mishra Countability Axioms