Real Analysis with Economic Applications - Chapter C pps

Chapter C M etric Spaces This chapter provides a self-contained review of the basic theory of m etric spaces. Chances are good that y ou are familiar with the rudiments of this theory, so our exposition starts a b it faster than us ual. But don’t wo rry, we slow do wn wh en we get to the “real stuff,” that being the analy sis of the properties of connec ted n e ss , separab ility, compactness and completeness for metric spaces. Con n ecte dne s s is a ge ometric propert y th a t w ill be of limited u se in this course. Consequently, its discussion here is quite b rief; all we do is to iden tify the connected subsets of R, and p repare fo r t he Intermediate Value T heorem that will be given in the next chapt er. Our treatme nt o f separability is also r e latively short, even though this c o n cep t will be important for us later on. Because s e p arab ility u s ua lly makes an appearance only in relatively advan c ed contexts, we will study th is propert y in greater detail later. Utility theory that w e sketc he d out in Section B.4 can be tak e n to the next le vel withthehelpofevenanelementaryinvestigationofconnectedandseparablemetric spaces. As a brief application, therefore, we formulate here the “metric” v ersions of some of the utility represen tation theorems that w ere p rov e d in that section. The story will be brought to its conclusion in Chapter D. The bulk of this chapter i s de voted to t he analysis of met ric spaces t hat a re e ither compact or c om p lete. A good understanding of these two properties is essen tial for real analysis and optimization theory, so w e spend quite a bit of time studying them. In particular, we consider several examples, give t wo proofs of the Heine- Borel Theorem for good measure, and discuss wh y closed and bounded spaces need not be compact in general. Totally bounded sets, the sequential characterization of com pactnes s, and the rela t i o nship between compactne ss and com pleteness, are also studied with care. Most of the results established in this chapter are relatively preliminary observa- tions whose main purpose is to create good grounds to deriv e a n umber of “ deeper” facts in later chapters. But there is one major exception: the Banach Fixed Poin t Theorem . While elementary, and has an amazingly simple proof, this result is of sub- stantial in tere st, and has nu merous applica tion s. We th u s exp lore it here at len gth . In p articular, w e consider so me of th e variants of this celebr ated theorem, an d sho w how it can be used to prov e the “existence” of a solution to certain t ypes of func- tional equ ation s. As a major application, we prove here both th e local a nd global versions of the fun da me ntal existe nce theorem of Emile Picard for differential e qua- tions. Two m ajor generalizations of the Ban ach Fixed P o int Theorem , along w ith further applications, will be considered in subsequen t chapters. 1 1 Among the excellent introductory references for the analysis of metric spaces are Sutherland 87 1BasicNotions Recall that w e think of a real function f on R as con t inuous at a given point a ∈ R iff the image of a poin t (under f)whichisclosetoa is itse lf close to f(a). So, for instance, the indicator function 1 { 1 2 } on R is not continuous a t 1 2 , because points that are arbitrarily close t o 1 2 are not mapped by th is fu nc tion to points tha t are arbitrarily closetoitsvalueat 1 2 . On the other hand, this function is continuous at every other point in its domain . It is crucial to un derstand a t the outset that this “geom etr ic” wa y of thinking about continuity depends intrinsically on the “distance” bet ween two points on the real line. While there is an obvious measur e of distance in R, this observation is importan t precisely because it paves way towards thinking about the contin u ity of function s defined on more complicated sets on whic h the meaning of the te rm “close” is not transparen t. As a prerequisite for a suitably general analysis of continuous functions, therefore, we need to elaborate on the notion of d istance between two elements of an arbitrary set. This is precisely what we intend to d o in this section. 1.1 M etric Spaces: Definitions and Examples We begin with the for mal definition of a metric space. Dhilqlwlrq. Let X beanynonemptyset.Afunctiond : X × X → R + that satisfies the following properties is called a distance function (or a metric)onX:Forany x, y, z ∈ X, (i) d(x, y)=0if an d only if x = y, (ii) (Symmetry) d(x, y)=d(y, x), (iii) (Triangle Inequality ) d(x, y) ≤ d(x, z)+d(z, y). If d is a d istance func tion on X, we say that (X, d) is a metric space,andreferto the elements of X as points in (X, d). If d satisfie s (ii) and (iii), and d(x, x)=0for an y x ∈ X, then w e say that d is a semim etric on X, and (X, d) is a semimetric space. (1975), Rudin (1976), Kaplansky (1977), and Haaser and Sulliva n (1991). O f the more recent expositions, my personal favorite is Carothers (2000). The firstpartofthatbeautifullywritten book not only pro vides a much broader perspective of metric spaces than I am able to do here, but it a lso cov ers additional topics (s uch as compactificationandcompletionofmetricspaces,and category-type theorems), and sheds light to the historica l dev elopmen t of the material. For a more advanced (but still very readable) account, I should refer you to Royden (1994), which is a classic text on real analysis. 88 Rec all that we th in k of the distance bet wee n t wo points x and y on the real line as |x − y| . Thus the map (x, y) → |x −y| serves as a function that tells us how muc h apart are any two elements of R from each other. Among others, this function satisfies properties (i)-(iii) of the definition above (Exam p le A.7). By wa y of abstraction , the notion of distance function is built only on these three properties. It is remarkable that these properties are strong enough to introduce to an arbitrary nonem pty set a geometry ric h enough to build a satisfactory theory of continuous functions. 2 Notation. When th e (semi)metric und er c onsider ation is ap pa rent from the context, it is customa ry to dispense with the notation (X, d), and refer to X as a metric space. We also adhere to th is convention here (and spare the notation d for a generic metric on X). But when we feel that there is a danger of confusion, or we endo w X with a particular metric d, t h en we shall reve rt bac k to the more d escriptive notation (X, d). Let us look at some standard examples of metric spaces. E{dpsoh 1. [1] Let X be any nonem pty set. A trivial w ay of making X ametric spaceistousethemetricd : X × X → R + whic h is defined by d(x, y):=  1,x= y 0,x= y . It is easy to chec k that (X, d) is indeed a metric space. Here d is called t h e discrete metric on X, and (X, d) is called a discrete space. [2] Let X := {x ∈ R 2 : x 2 1 + x 2 2 =1}, and define d ∈ R X×X by letting d(x, y) be thelengthoftheshorterarcinX that join x and y. It is easy to see that this defines d as a metric on X, and thus (X, d) is a m etric spac e. [3] Given any n ∈ N, there are various ways of metrizing R n . Indeed, (R n ,d p ) is a m etric space for eac h 1 ≤ p ≤∞, where d p : R n × R n → R + is defined by d p (x, y):=  n  i=1 |x i − y i | p  1 p for 1 ≤ p<∞, and d p (x, y):=max{|x i − y i | : i =1, , n} for p = ∞. It is easy to see that each d p sat i sfies the first two axioms of being a distan c e func tion . The verification of the triangle inequalit y in the case of p ∈ [1, ∞) is, on the other 2 The c oncept of metric space was first introduced in the 1906 dissertation of Maurice Fréchet (1878-1973). (We owe the term “metric space” to Felix Hausdorff, however.) Considered as one of the major founders of modern real (and functional) analysis, Fréchet is also the mathematician who first introduced the abstract formulation of compactness and completeness properties. (See Dieudonné (1981) and Taylor (1982)). 89 hand, not a trivial matter. It rath e r follows from the followin g celebr ate d result of Herman n Minkowski: Min kowsk i’s Ineq u ality 1. For any n ∈ R, a i ,b i ∈ R,i=1, , n, and any 1 ≤ p< ∞,  n  i=1 |a i + b i | p  1 p ≤  n  i=1 |a i | p  1 p +  n  i=1 |b i | p  1 p . To be able to move faster, we postpone the p r oof o f this importan t inequ ality to t he end of this subsection . You are invited at this point, how ever, to sho w that (R n ,d p ) is not a m etric space for p<1. It may be instructive to examine the geometry of the unit “circle” C p := {x ∈ R 2 : d p (0,x)=1} for various choice s of p. (Here 0 stands for the 2-vector (0, 0).)ThisisdoneinFigure 1, which suggests that the sets C p in some sense “converges” to the set C ∞ . Indeed, for eve r y x, y ∈ R 2 , we have d m (x, y) → d ∞ (x, y). (Proof?) ∗∗∗∗FIGURE C.1 ABOUT HERE ∗∗∗∗ The space (R n ,d 2 ) is ca lled the n-dimensional Euclidean space in analysis. When we refer to R n in th e sequel without specifying a particular metric, you sh ou ld understand that w e view th is set as metrized by the metric d 2 . That is to say, the notation R n is spared fo r the n-dimensional Euclidean space in what follows. If we wish to endow R n with a metric d ifferent than d 2 , w e will be explicit about it. Notation. Throughout this text we denote the metric space (R n ,d p ) as R n,p for a ny 1 ≤ p ≤∞. Ho wever, alm ost always, w e use the notatio n R n inst ead o f R n,2 . Before leaving this ex a mple, let’s see h ow w e can metrize an extended Euclide an space, say R. For this purpose, w e define the function f : R → [−1, 1] by f(−∞):= −1,f(∞):=1, and f(x):= x 1+|x| for all x ∈ R. The standa rd metric d ∗ on R is the n defined by d ∗ (x, y):=|f(x) − f(y)| . The im portant thing to obser ve he re is that this makes R ametricspacewhichis essentially identical to [−1, 1]. This is because f is a biject ion from R onto [−1, 1] that leaves the distance between any two points in tact: d ∗ (x, y)=d 1 (f(x),f(y)) for an y x, y ∈ R. So, w e should expect that an y metric property that is true in [−1, 1] is also true in R. 3 3 This point may be somewhat v ague rig ht now. That’s okay, i t will become clearer bit by bit as we move on. 90 [4] For any 1 ≤ p<∞, we define  p :=  (x m ) ∈ R ∞ : ∞  i=1 |x i | p < ∞  . This set is metrized b y means of the metric d p :  p ×  p → R + with d p ((x m ) , (y m )) :=  ∞  i=1 |x i − y i | p  1 p . (When we speak of  p as a metric space, we always have this metric in m in d !) O f course,wehavetocheckthatd p is well-defined as a real-valued function, and that it satisfies the triang le inequ ality. But no w orries, these facts follow r e ad ily from th e following generalization of Mink owsk i’s Inequality 1: Min kowsk i’s Ine q ua lity 2. For any (x m ) , (y m ) ∈ R ∞ and 1 ≤ p<∞,  ∞  i=1 |x i + y i | p  1 p ≤  ∞  i=1 |x i | p  1 p +  ∞  i=1 |y i | p  1 p . (1) We will pro ve this inequality at the end of this subsection. You should assume its validity for now, and v e r ify that d p is a m etric on  p for any 1 ≤ p<∞. By  ∞ , we mean the s et of all bounded real sequences, th at is,  ∞ := {(x m ) ∈ R ∞ :sup{|x m | : m ∈ N} < ∞} . It is implicitly understood that this set i s e ndowed with the metric d ∞ :  ∞ × ∞ → R + with d ∞ ((x m ) , (y m )) := sup{|x m − y m | : m ∈ N}. That d ∞ is indeed a metric will be verified below. This m etric is called the sup- metric on the set of all bounded r eal sequences. Before we leave th is example let us stress that a ny  p space is smaller than the set of all r eal sequences R ∞ since the mem bers of such a space are real sequences that are either bounde d or that satisfy some form of a summability condition (that ensures that d p is real-valued). Indeed, no d p defines a distance function on the en tire R ∞ . (Why?) But this does not m ean that we cannot metrize th e set of all real sequences in a us e fu l wa y. We c an, and we will, later in th is c h a p ter. [5] Let T beanynonemptyset. By B(T ) we mean th e set o f all bounded real function s defined on T, that is, B(T ):=  f ∈ R T :sup{|f(x)| : x ∈ T } < ∞  . We will always think of this space as metrized by the sup-metric d ∞ : B(T) × B(T ) → R + whic h is defined by d ∞ (f,g):=sup{|f(x) −g(x)| : x ∈ T }. 91 It is easy to see that d ∞ is real-valued. Indeed, for any f, g ∈ B(T), d ∞ (f,g) ≤ sup{|f(x)| : x ∈ T } +sup{|g(x)| : x ∈ T} < ∞. It is also readily checked that d ∞ satisfies the first two req uirem ents o f being a distance function. As for the triangle inequalit y, all we need is to invoke the cor responding property of the absolute value function (Example A.7). After all, if f,g,h ∈ B(T ), then |f(x) − g(x)| ≤ |f(x) − h(x)| + |h(x) −g(x)| ≤ sup{|f(y) −h(y)| : y ∈ T} +sup{|h(y) −g(y)| : y ∈ T } = d ∞ (f,h)+d ∞ (h, g) for any x ∈ T, so d ∞ (f,g)=sup{|f(x) − g(x)| : x ∈ T} ≤ d ∞ (f,h)+d ∞ (h, g) . Given that a sequence and/or an n-vector can alwa ys be thought of as special functions (Se c tion A.1 .6), it is plain that B({1, , n}) co inc i d es w ith R n,∞ (for any n ∈ N) while B(N) coincides w ith  ∞ . (Right?) Th e ref ore , the inequality we just established pro ves in one strok e that both R n,∞ and  ∞ are metric spaces.  Remark 1. Distance functions need not bounded. However, giv en any metric space (X, d), we can always find a bounded metric d  on X that orders th e distances between points in the space ordinally the same way as the original metric. (That is, such a metric d  satisfies: d(x, y) ≥ d(z, w) iff d  (x, y) ≥ d  (z,w) for all x, y, z, w ∈ X.) Indeed, d  := d 1+d is such a d istance function. (Note. We have 0 ≤ d  ≤ 1.) As we proceed further, it will becom e clear that there is a good sense in which (X, d) and (X, d 1+d ) c an be though t of as “equivale nt” in terms of certain characteristics (and not so in terms of others). 4  If X i s a metric space (with metric d )and∅ = Y ⊂ X, we can view Y as a metric space in its own righ t by using the distance function induced b y d on Y. More precisely, we mak e Y a metric space by means of the distance function d| Y ×Y . We then say that (Y,d| Y ×Y ),orsimplyY, is a metric subspace of X. Fo r instance, we think of a ny in terval, say [0, 1], as a metric subspace of R; this means simply that the distance between any two elements x and y of [0, 1] is calculate d by viewing x and y as points in R: d 1 (x, y)=|x − y| . Of course, we can also think of [0, 1] as a 4 This is definitely a good point to keep in mind. When there is a “natural” unbounded metric on the space that you are working with, but for some reason you need a bounded metric, you can alw ays modify the original metric to get a new bounded and ordinally equivalent metric on your space. (More on this in Section 1.5 below.) Sometimes, and we will encounter such an instance later, this little trick does wonders. 92 metric subspace of R 2 . Formally, we would d o this by “identifying” [0, 1] with th e set [0, 1] ×{0} (or wit h {0}×[0, 1], or with [0, 1] ×{47}, etc.), and consider [0, 1] ×{0} as a metric subspace of R 2 . This would render the distance between x and y equal to, again, |x − y| (for, d 2 ((x, 0), (y, 0)) = |x −y|). 5 Con v ention. Throughout this book, when we consider a nonempty subset S of a Euclidean space R n as a metric spac e without e xp lic itly men t ion ing a particula r metric, you should understand that we view S a m etric subspace of R n . E{dpsoh 2. [1] For any positiv e integ er n, we may think of R n as a m etric subspace of R n+1 by identify ing it with th e subset R n ×{0} of R n+1 . By induction, therefore, R n can be though t of as a m etric subspace of R m for any m, n ∈ N with m>n. [2] Let −∞ <a<b<∞, an d consider t he metric space B[a, b] introduced in Example 1.[5]. R ecall that ev e ry contin uous function on [a, b] is bounded (Exerc ise A.53), and h ence C[a, b] ⊆ B[a, b]. Consequently, w e can consider C[a, b] as a metric subspace of B[a, b]. Indeed, throughout this text, whenever w e talk about C[a, b] as a metric space, we think of the distance bet ween any f and g in C[a, b] as d ∞ (f,g), unless otherwise is explic itly mention ed. [3] Let −∞ <a<b<∞, and recall that we denote the set of all con tin uously differentiable functions on [a, b] by C 1 [a, b] (Sectio n A .4.2). The m e tric tha t is used for this space is usually not the sup-metric. That is, we do not define C 1 [a, b] as a metric subspace of B[a, b]. (There are a good reasons for this, but w e’ll get to them later.) Instea d, C 1 [a, b] is commonly metrized by means of the distance function D ∞ : C 1 [a, b] × C 1 [a, b] → R + define d by D ∞ (f,g):=d ∞ (f,g)+d ∞ (f  ,g  ). It is th is metric that we have in mind w hen talking about C 1 [a, b] as a metric space.  Exercise 1. H If (X, d) and (X, ρ) aremetricspaces,is(X, max{d, ρ}) necessarily a metric space? Ho w about (X, min{d, ρ})? Exercise 2. For any metric space X, show that |d(x, y) −d(y, z)| ≤ d(x, z) for all x, y, z ∈ X. Exercise 3. For an y semimetric space X,define the binary relation ≈ on X by x ≈ y iff d(x, y)=0. Now define [x]:={y ∈ X : x ≈ y} for all x ∈ X, and let X := {[x]:x ∈ X}. Finally, define D : X 2 → R + by D([x], [y]) = d(x, y). (a) Show that ≈ is an equivalence relation on X. (b)Provethat(X ,D) is a metric space. 5 Quiz. Is the metric space g iven in Example 1.[2] a metric subspace of R 2 ? 93 Exercise 4. Show that (C 1 [0, 1],D ∞ ) is a metric space. Exercise 5. Let (X, d) be a metric space and f : R + → R a concave and strictly increasing function with f(0) = 0. Show that (X, f ◦ d) is a metric space. The final orde r of busin es s in this sub s ection is to pr ove Minkowski’s Inequalities whichwehaveinvokedabovetoverifythatR n,p and  p aremetricspacesforany 1 ≤ p<∞. Since the first one is a special case of the second (yes?), all we need is to establish Minkowsk i’s Inequa lity 2. Proof of Minkow ski’s Inequality 2. Take any (x m ) , (y m ) ∈ R ∞ and fixany1 ≤ p<∞. If either  ∞ |x i | p = ∞ or  ∞ |y i | p = ∞, then (1) becomes trivial, so we assume that  ∞ |x i | p < ∞ and  ∞ |y i | p < ∞. (1) is also trivially true if either (x m ) or (y m ) equals (0, 0, ), so we focus on the case where both α := (  ∞ |x i | p ) 1 p and β := (  ∞ |y i | p ) 1 p are positiv e real nu mbers. Define the real sequences (ˆx m ) or (ˆy m ) by ˆx m := 1 α |x m | an d ˆy m := 1 β |y m | . (Notice that  ∞ |ˆx i | p =1=  ∞ |ˆy i | p .) Using the triangle inequality for the absolute value function (Examp le A.7), and the fact that t → t p is an inc re a s ing map on R + , we find |x i + y i | p ≤ (|x i | + |y i |) p =(α |ˆx i | + β |ˆy i |) p =(α + β) p  α α+β |ˆx i | + β α+β |ˆy i |  p for eac h i =1, 2 , But since t → t p is a convex m a p on R + , we have  α α+β |ˆx i | + β α+β |ˆy i |  p ≤ α α+β |ˆx i | p + β α+β |ˆy i | p ,i=1, 2, , and hence |x i + y i | p ≤ (α + β) p  α α+β |ˆx i | p + β α+β |ˆy i | p  ,i=1, 2, Summing over i, th en, ∞  i=1 |x i + y i | p ≤ (α + β) p  α α+β ∞  i=1 |ˆx i | p + β α+β ∞  i=1 |ˆy i | p  =(α + β) p  α α+β + β α+β  1 p . Thus  ∞ |x i + y i | p ≤ (α + β) p which is equ ivalen t to (1).  We conclude by noting that (1) holds as an equalit y (for an y given 1 <p<∞) iff either x =(0, 0, ) or y = λx for som e λ ≥ 0. The proof is left as an exercise. 94 1.2 Open and Closed Sets We no w review a n umber of fundamental concepts regarding metric spaces. Dhilqlwlrq. Let X be a metric (or a sem imetric) space. Fo r an y x ∈ X and ε > 0, we define the ε-neigh borhood of x in X as the set N ε,X (x):={y ∈ X : d(x, y) < ε}. In turn, a neighborhood of x in X is any subset of X that con tains at least one ε-neighborhood of x in X. The first thing that you should note about the ε-neighborhoo d of a point x in a ( s e m i)metric sp ace is that such a set is n ever empty, for it contains x. Secondly, mak e sure y ou understand that this notion is based on four primitive s. Ob v iou sly, the ε-neighborhood o f x in a metric s pace X depends on ε and x. But it also depends on the set X and the distance function d used to metrize this set. For instance, the 1-ne ighborhood of 0 :=(0, 0) in R 2 is {(x 1 ,x 2 ) ∈ R 2 : x 2 1 + x 2 2 < 1}, whereas the 1-ne ighborhood of 0 in R ×{0} (vie wed as a m e tric subspace of R 2 )is{(x 1 , 0) ∈ R 2 : −1 <x 1 < 1}. Similarly, the 1-neighborhood of 0 in R 2 is d istin c t from that in R 2,p for p =2. The notion of ε-neighborhoods pla ys a m ajor role in real analysis mainly through the following definition. Dhilqlwlrq. A subset S of X is said to be open in X (or an open subset of X) if, for each x ∈ S, th ere exists an ε > 0 suc h that N ε,X (x) ⊆ S. AsubsetS of X is said to be closed in X (or a closed subset of X)ifX\S is open in X. Because an ε-neighborhood of a point is inheren tly connected to the underlying metric space, so does the notions of open and c losed sets. P lease keep in mind that c han ging the metric on a giv e n set, or concentrating on a metric subspace of the original metric space, would in general yield differentclassesofopen(andhence closed) sets. Dhilqlwlrq. Let X be a metric space and S ⊆ X. The largest open set in X that is contained in S (th at is, the ⊇-m ax imum of the class of all open subsets of X contained in S) is ca lled the inte rior of S (r e lative to X), and is denoted by int X (S). On the other hand, the closure of S (relative to X), denoted by cl X (S), is defined as the sma llest closed set in X that contains S (that is, the ⊇-minim um of the class of all closed subsets of X that con t ain S). The boundary of S (relative to X), denoted by bd X (S), is de fined as bd X (S):=cl X (S)\int X (S). 95 Let X be a metric s pace, and Y a metric subspace of X. Fo r any subset S of Y, we ma y think o f the interior of S as lying i n X or in Y. (And y e s , these may well be quite different!) It is for th is reason t ha t we use the notation int X (S), instead of int(S), to mean the interior of S relative to the m etric spa ce X. However, if there is only one metric space under consideration, or the context lea ves no room for confusion, w e may, and will, simply write int(S) to denote the interior of S relative to the appropriate space. (The sam e comments a p p l y to the closure and boun d ary operators as well.) E{dpsoh 3. [1] In a ny metric space X, the sets X and ∅ are both open and closed. (The sets which a re both open and closed are sometim es called clopen in analysis.) On the o ther hand, for any x ∈ X and ε > 0, the set N ε,X (x) is open, and the set {x} is closed. To prov e that N ε,X (x) is open, t ake any y ∈ N ε,X (x) and d efine δ := ε−d(x, y) > 0. We have N δ,X (y) ⊆ N ε,X (x) because, by the triangle inequalit y, d(x, z) ≤ d(x, y)+d(y, z) <d(x, y)+ε − d(x, y)=ε for an y z ∈ N δ,X (y). (See Figure 2 to see the intuition of the argument.) To prove that {x} is closed, we need to show that X\{x} is open. If X = {x}, there is nothing to p rove. (Yes?) On the o ther ha nd , if there exists a y ∈ X\{x }, we then have N ε,X (y) ⊆ X\{x}, where ε := d(x, y). It follows that X\{x} is open, and {x} is closed. ∗∗∗∗FIGURE C.2 ABOUT HERE ∗∗∗∗ [2] Any subset S of a nonem p ty set X is open with respect to the discrete metric. For, if x ∈ S ⊆ X, then w e ha ve N 1 2 ,X (x)={x} ⊆ S, where the discrete m etric is used in computing N 1 2 ,X (x). Th us: A ny subset of a discrete space is clopen. [3] It is possible for a set in a metric sp ac e to be neither open nor closed . In R, for instance, (0, 1) is open, [0, 1] is clos ed, and [0, 1) is neither open n or closed. But observe that the structure of the mother metric space is crucial for the v alidity of these statements. For instance, the set [0, 1) is open when considered as a set in themetricspaceR + . (Indeed, re lative to this metric subspa ce of R, 0 belongs to the in terior of [0, 1), and the boundary of [0, 1) equals {1}.) More ge n erally, th e follo wing fact is tru e : Exercise 6 . Given any metric space X, let Y beametricsubspaceofX,andtake any S ⊆ Y. Show that S is open in Y iff S = O ∩ Y forsomeopensubsetO of X, anditisclosedinY iff S = C ∩Y forsomeclosedsubsetC of X. Warnin g. Giv en an y m etric space X, let Y beametricsubspaceofX, and take any U ⊆ Y. An immediate application of Exe rcis e 6 shows that U is open in X on ly if U is open in Y. 96 [...]... Show that a separable metric space need not be compact, but a compact metric space X is separable (b) Show that a connected metric space need not be compact, and a compact metric space need not be connected 3.2 Compactness as a Finite Structure What is compactness good for? Well, the basic idea is that compactness is some sort of a generalization of the notion of finiteness To clarify what we mean by this,... (Section D.2) Exercise 16 Show that the closure of any connected subset of a metric space is connected Exercise 17 Show that if S is a finite (nonempty) class of connected subsets of a metric space such that S = ∅, then S must be connected Exercise 18 For any given n ∈ {2, 3, }, prove that a convex subset of Rn is necessarily connected, but not conversely 104 Exercise 19.H Show that a metric space which... denote a generic sequence in a given (abstract) metric space X by (xm ), (ym ) etc (This convention becomes particularly useful when, for instance, the terms of (xm ) are themselves sequences.) The generic real (or extended real) sequences are denoted as (xm ), (ym ), etc., and generic sequences of real functions are denoted as (fm ), (gm ), etc 10 I could write the argument more compactly as d(x, y)... (Exercise 6) while O ∪ U = I, contradicting the connectedness of I Conclusion: A nonempty subset of R is connected iff it is an interval We won’t elaborate on the importance of the notion of connectedness just as yet This is best seen when one considers the properties of continuous functions defined on connected metric spaces, and thus we relegate further discussion of connectedness to the next chapter. .. O and U of I Pick any a ∈ O and b ∈ U and let a < b without loss of generality Define c := sup{t ∈ O : t < b}, and note that a ≤ c ≤ b and hence c ∈ I (because I is an interval) If c ∈ O, then c = b (since O ∩ U = ∅), so c < b But, since O is open, there exists an ε > 0 such that b > c + ε ∈ O, which contradicts the choice of c (Why?) If, on the other hand, c ∈ O, then a < c ∈ U (since I = O ∪ U ) Then,... ε > 0 such that S ⊆ Nε,Rn (x) for some x ∈ S Therefore, S must be a closed subset of a cube [a, b]n.32 But [a, b]n is compact by the Heine-Borel Theorem, and hence S must be compact by Proposition 4 A common mistake is to “think of” any closed and bounded set in a metric space as compact This is mostly due to the fact that some textbooks which focus exclusively on Euclidean spaces define compactness... for a subset of C[ 0, 1] which is not equicontinuous 4 Sequential Compactness Recall that we can characterize the closedness property in terms of convergent sequences (Proposition 1) Given that every compact space is closed, it makes sense to ask if it is possible to characterize compactness in the same terms as well The answer turns out to be affirmative, although the associated characterization is more... 1, ) is not a Cauchy sequence in R, for |(−1)m − (−1)m+1 | = 2 for all m ∈ N Warning For any sequence (xm ) in a metric space X, the condition that consecutive terms of the sequence get closer and closer, that is, d(xm , xm+1 ) → 0, is a necessary but not sufficient condition for (xm ) to be Cauchy The proof of the first claim is easy To verify the second claim, consider the real sequence (xm ) where... divergent real 1 sequence but ln(m + 1) − ln m = ln(1 + m ) → 0 121 Exercise 37 Prove: If (xm ) is a sequence in a metric space X such that < ∞, then it is Cauchy ∞ d(xi , xi+1 ) The first thing to note about Cauchy sequences is that they are bounded That is, the set of all terms of a Cauchy sequence in a metric space is a bounded subset of that space Indeed, if (xm ) is a Cauchy sequence in a metric space... a sequence (Cm ) of cubes in Rn such that, for each m ∈ N, we have (i) Cm+1 ⊂ Cm , (ii) the length of an edge of Cm is 21 (b−a), and (iii) Cm is not covered m by any finite subset of O (b) Use part (a) to prove the Heine-Borel Theorem The following simple fact helps us find other examples of compact sets Proposition 4 Any closed subset of a compact metric space X is compact Proof Let S be a closed subset . Chapter C M etric Spaces This chapter provides a self-contained review of the basic theory of m etric spaces. Chances are good that y ou are familiar with the rudiments of. vides a much broader perspective of metric spaces than I am able to do here, but it a lso cov ers additional topics (s uch as compactificationandcompletionofmetricspaces,and category-type theorems),. XO. 1.3 Conv ergent Sequences The notion of closedness (and hence openness) of a set in a metric space can be c haracterized by means of the sequences th at live i n that space. Since th is char acter- ization