Week 6: Topology & Real Analysis Notes

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1 Week 6: Topology & Real Analysis Notes To this point, we have covered Calculus I, Calculus II, Calculus III, Differential Equations, Linear Algebra, Complex Analysis and Abstract Algebra. These topics probably comprise more than 90% of the GRE math subject exam. The remainder of the exam is comprised of a seemingly random selection of problems from a variety of different fields (topology, real analysis, probability, combinatorics, discrete math, graph theory, algorithms, etc.). We can t hope to cover all of this, but we will state some relevant definitions and theorems in Topology and Real Analysis. Topology The field of topology is concerned with the shape of spaces and their behavior under continuous transformations. Properties regarding shape and continuity are phrased using the concept of open sets. Definition 1 (Topology / Open Sets). Let X be a set and τ be a collection of subsets of X. We say that τ is a topology on X if the following three properties hold: (i), X τ (ii) If T 1,... T n is a finite collection of members of τ, then (iii) If {T i } i I is any collection of members of τ, then i I T i τ n T i τ i=1 In this case, we call the pair (X, τ) a topological space and we call the sets T τ open sets. Note, there are two topologies which we can always place on any set X: the trivial topology τ = {, X} and the discrete topology τ = P(X). Having defined open sets, we are able to define closed sets. Definition 2 (Closed Sets). Let (X, τ) be a topological space. A set S X is called closed iff S c τ. That is, S is defined to be closed if S c is open. The words open and closed can be a bit confusing here. Often times students mistakenly assume that a set is either open or closed; that these terms are mutually exclusive and describe all sets. This is not the case. Indeed, sets can be open, closed, neither open nor closed, or both open and closed. In any topological space (X, τ), the sets and X are both open and closed. By De Morgan s laws, since finite intersections and arbitrary unions of open sets are open, we see that finite unions and arbitrary intersections of closed sets remain closed. Example 3. The set of real numbers R becomes a topological space with open sets defined as follows. Define to be open and define T R to be open iff for all x T, there 1

2 exists ε > 0 such that (x ε, x + ε) T. Prototypical open sets in this topology are the open intervals (a, b) = {x R : a < x < b}. Indeed, this interval is open because for x (a, b), we can take ε = min{ x a, x b } and we will find that (x ε, x + ε) (a, b). We can combine open sets via unions or (finite) intersections to make more open sets; for example (0, 1) (3, 5) is also an open set. Likewise, prototypical closed sets are closed intervals [a, b] = {x R : a x b}, and any intersection or (finite) union of such sets will remain closed. As was observed above and R are both open and closed; in fact, in this space, these are the only sets which are both open and closed, though it is easy to construct sets which are neither open nor closed. Consider the set [0, 1) = {x R : 0 x < 1}. This set is not open because the point 0 is in the set, but it cannot be surrounded by an interval which remains in the set. The complement of this set is (, 0) [1, ). This set is not open since 1 is in the set but cannot be surrounded by an interval which remains in the set. Since the complement is not open, the set [0, 1) is not closed. Note, this topology is called the standard topology on R. Example 4. While the above example defines the standard topology on R, it is easy to come up with non-standard topologies as well. Indeed, let us now define T R to be open if T can be written as a union of sets of the form [a, b) = {x R : a b < x}. These open sets comprise a topology on R. In this topology a prototypical open set is of the form [a, b). What other sets are open in this topology? Notice that (a, b) = [a + 1/n, b) n=1 which shows that sets of the form (a, b) remain open in this topology. Also notice that since [a, b) is open, we define [a, b) c = (, a) [b, ) to be closed. However, both (, a) and [b, ) are easily seen to be open, so the set (, a) [b, ) is also open as a union of open sets. Since this set is open, it s complement [a, b) is closed. Hence in this topology, all sets of the form [a, b) are both open and closed. The intervals [a, b] are closed and not open in this topology. Note, this topology is called the lower limit topology on R. Notice in these example, the lower limit topology contains as open sets all of the sets which are open in the standard topology. In this way, the lower limit topology has more open sets and we can think of the lower limit topology containing the standard topology. We define these notions here. Definition 5 (Finer & Coarser Topologies). Suppose that X is a set and τ, σ are two topologies on X. If τ σ, we say that τ is coarser than σ and that σ is finer than τ. On any space X, the finest topology is the discrete topology P(X) and the coarsest is the trivial topology {, X}. A finer topology is one that can more specifically distinguish between elements. 2

3 Definition 6 (Interior & Closure). Let (X, τ) be a topological space and let T X. The interior of T is defined to be the largest open set contained in T. The closure of T is defined to be the smallest closed set containing T. We denote these by int(t ) and cl(t ) respectively. In symbols, we have int(t ) = S and cl(t ) = S. S τ,s T S c τ,t S Other common notations are T for the interior of T, and T for the closure of T. Example 7. Considering R with the standard topology and a, b R, a < b, we have int([a, b)) = (a, b) and cl([a, b)) = [a, b]. In both of the examples above, there was some notion of a prototypical open set, from which other open sets can be built. We give this notion a precise meaning here. Definition 8 (Basis (Base) for a Topology). of subsets of X such that Let X be a set and let β be a collection (1) X = B β B, (2) if B 1, B 2 β, then for each x B 1 B 2, there is B 3 β such that x B 3 and B 3 B 1 B 2. Then the collection of sets τ = { T : T = i I B i for some collection of sets {B i } i I β } forms a topology on X. We call this τ the topology generated by β, and we call β a basis for the topology τ. This is half definition and half theorem: we are defining what it means to be a basis, and asserting that the topology generated by a basis is indeed a topology. If we can identify a basis for a topology, then the basis sets are the prototypical open sets, and all other open sets can be built as unions of the basis sets. Morally, basis sets are representatives for the open sets; if you can prove a given property for basis sets, the property will likely hold for all open sets. Often times it is easiest to define a topology by identifying a basis. Example 9. Above we defined the standard topology on R by saying that a set T is open if for all x T, there is ε > 0 such that (x ε, x + ε) T. It is important to see this definition of the topology; however, this is a much more analytic than topological definition. The topological way to define the standard topology on R would be to define it as the topology generated by the sets (a, b) where a, b R, a < b. Indeed, these two definitions of the standard topology are equivalent as the following proposition shows. 3

4 Proposition 10. Suppose that (X, τ) is a topological space and that β is a basis for the topology τ. Then T τ iff for all x T, there is B β such that x B and B T. It is important to identify when two bases in a topological space generate the same topology and this next proposition deals with that question. Proposition 11. Suppose that X is a set and β 1, β 2 are two bases for topologies τ 1 and τ 2. Then τ 1 τ 2 iff for every B 1 β 1, and for every x B 1, there is B 2 β 2 such that x B 2 and B 2 B 1. (Be very careful not to mix up the inclusions in this statement. What this is essentially saying is that β 2 generates a larger (finer) topology iff β 2 has more (smaller) sets.) Informally, the basis β 2 generates a finer topology if we can squeeze basis sets from β 2 inside basis sets from β 1 (and not only that, but we can construct basis sets from β 1 out of basis sets from β 2 ). Just as all groups have subgroups and all vector spaces have subspaces, there is a natural way to define subspaces of a topological space. Definition 12 (Subspace Topology). Let (X, τ) be a topological space and let Y X. Then the collection of sets σ = {Y T : T τ} forms a topology on Y. This topology σ is called the subspace topology on Y inherited from (X, τ). Again, this is part definition and part theorem; we are asserting that such σ does indeed define a topology on Y. Example 13. Consider [0, 3] R with the standard topology on R. Note that the subspace topology on [0, 3] includes standard open sets like (1, 2) since this set is open in R and (1, 2) = [0, 3] (1, 2). Now consider the set (1, 3]. This set is not open in R; however, it is open in the subspace topology on [0, 3], because (1, 4) is open in R and (1, 3] = [0, 3] (1, 4). Likewise there is a natural way to combine topological spaces in a Cartesian product. Definition 14 (Product Topology). Let (X, τ), (Y, σ). Recall the Cartesian product is given by coupling elements of X and Y : X Y.= {(x, y) : x X, y Y }. It is tempting to define a topology on X Y comprised of sets of the form T S for T τ, S σ. However, these do not form a topology on X Y since a union of sets of this form will not be of this form anymore. So rather, we let β = {T S : T τ, S σ} form the basis for a topology on X Y. The topology generated by β is denoted τ σ and the space (X Y, τ σ) is 4

5 the called the product space of X and Y. 1 Example 15. Consider R with the standary topology, which we call τ 1. The product space (R R, τ 1 τ 1 ) can be visualized by drawing the plane with standard open sets as rectangles (a, b) (c, d) = {(x, y) R R : a < x < b and c < y < d}. Alternatively, we can consider the set R 2 of 2-dimensional vectors. On this space, we consider the topology τ 2 generated by circles: B r (v) = {z R 2 : v z < r} for v R 2 and r > 0 (indeed, this is called the standard topology on R 2 ). We can identify each vector v = ( x y ) R 2 with the coordinates (x, y) R R. Since this is a bijective map, the sets R R and R 2 are really the same. We d like to know if the topologies τ 1 τ 2 and τ 2 are the same. To prove they are the same, consider the bases β 1 = {(a, b) (c, d) : a, b, c, d R, a < b, c < d} β 2 = { B r (v) : v = ( x y ) R 2, r > 0 }. and For any B 1 = (a, b) (c, d) β 1, take any (x, y) B 1 and let r = min{x a, b x, y x, d y}. Then for v = ( x y ), we will have x B r (v) B 1 ; this shows that for any (x, y) B 1, we can find a set B 2 β 2 such that (x, y) B 2 and B 2 B 1. Hence by Proposition 11, τ 1 τ 1 τ 2. Conversely, let v = ( x y ) and r > 0 and consider the set B 2 = B r (v) β 2. For any u = ( z w ) B r (v), define r = (r u v 2 )/ 2. Then the square B 1 = (z r, z + r ) (w r, w + r ) satisfies u = ( z w ) B 1 and B 1 B 2. Thus by Proposition 11, we have τ 2 τ 1 τ 1, and we can conclude that τ 2 = τ 1 τ 1. (Note, this inclusions of basis sets is pictured in Figure 1.) That is, the standard topology on R 2 is the product of two copies of the standard topology on R. More generally, for n N, we can define the standard topology on R n to be the topology generated by open balls and we will find that this is the same as the product of n copies of the standard topology on R. Topology gives us the minimum structure required to discuss limits and continuity. Indeed, in calculus we were only able to discuss these things because R is naturally a topological space with the standard topology. We give the topological definitions of limits and continuity here and discuss some of their properties. Definition 16 (Limit of a Sequence). Let (X, τ) be a topological space and let {x n } n=1 be a sequence of values in X. We say that {x n } n=1 converges to a limit x X if for any open set T τ such that x T, there is N N, such that x n T for all n N. We write this as x n x or lim n x n = x. 1 Note, this is only a good definition for the product topology when we are taking the product of a finite number of spaces. Indeed, if {(X i, τ i )} i I is an arbitrary collection of topological spaces, it is most natural to define the product topology on X = i I X i to be the coarsest topology so that the projection maps π i : X X i are continuous. The topology defined generated by sets of the form i I U i where U i τ i is then called the box topology. One can show that for a finite Cartesian product, the product topology and box topology agree with each other; this is not necessarily true for infinite products. (Another way to correctly define the product topology for an infinite product X = i I X i is to let it be generated by sets of the form i I U i where U i τ i and U i = X i for all but finitely many i I.) 5

6 Figure 1: A basis set from either topology τ 1 τ 1 or τ 2 can be fit around any point in a basis set from the other topology. Note that this is a generalization of the definition we gave for a limit in calculus; in calculus we are always using the standard topology on R. One feature of the limit in calculus is that limits are unique: if x n x and x n y, then x = y. This is not true in a general topological space. Example 17. Consider a space X with the trivial topology τ = {, X}. Take any sequence {x n } n=1 in X and apply the definition of the limit. For any x X, and any N N, we see that if T τ and x T, then T = X and x n T for all n N. Thus in this space, every sequence converges to every point. There are non-trivial topologies where limits are still non-unique, but our intution tells us that limits should be unique and we can add a simple property to ensure that they are. Definition 18 (Hausdorff Space). A topological space (X, τ) is called a Hausdorff space (or is said to have the Hausdorff property), if for all x, y X with x y, there are open sets T x, T y τ such that x T x, y T y and T x T y =. Proposition 19. Limits in Hausdorff spaces are unique. That is, if (X, τ) is a Hausdorff space and {x n } n=1 is a sequence in X, then {x n } n=1 can have at most one limit x X. Now we would like to discuss maps between spaces. As with linear transforms in linear algebra and homomorphisms in abstract algebra, we restrict our discussion to maps which preserve some of the underlying structure of the space. In topology, these are the continuous functions. Definition 20 (Continuous Function). Let (X, τ) and (Y, σ) be two topological spaces. A function f : X Y is said to be continuous iff f 1 (V ) = {x X : f(x) V } τ whenever V σ. That is, f is continuous if the preimage of every open set in Y is open in X. 6

7 This gives us a way to define continuity without ever considering individual points. Contrast this with the calculus definition of continuity where we first define what it means for a function to be continuous at a point, and then define continuous functions on a domain to be those which are continuous at each point. Of course, topology still has a notion of what it means to be continuous at a point. Definition 21 (Pointwise Continuity). Let (X, τ) and (Y, σ) be two topological spaces and let f : X Y. We say that f is continuous at the point x X, if for all V σ such that x V, we have f 1 (V ) τ. It is a good exercise to prove that when both X, Y are R with the standard topology, this definition of continuity is equivalent to the ε-δ definition of continuity presented in calculus. Continuity is an important property because it preserves certain features of topological spaces. Indeed, if f : X Y, we define the image of X under f by f(x) = {f(x) : x X}. We define σ f to be the collection of subsets V f(x) such that f 1 (V ) is open in X. If f is continuous, this collection will form a topology, and this topology σ f on f(x) will be the same as the subspace topology that f(x) inherits from (Y, σ). This shows that continuous functions respect the structure of the underlying spaces; in other words, continuous functions are the morphisms in the category of topological spaces. Above we showed that the spaces (R 2, τ 2 ) and (R R, τ 1 τ 1 ) are essentially the same space; it is important to be able to be able to identify equivalent spaces or distinguish between distinct topological spaces and continuity helps us do that. We define a few more properties of topological spaces here. Definition 22 (Connectedness). Let (X, τ) be a topological space and let C X. We say that the set C is disconnected iff there exist non-empty A, B τ such that C A B and A B =. We say the space is connected iff it is not disconnected. Example 23. Intuitively, a set is connected if it is in one whole piece; disconnected sets have separate pieces broken off from each other. Thus for example, in R with the standard topology, the set [5, 9) is connected while the set {1} [5, 9) (10, ) is disconnected. However note, this is only a heuristic! It works very well in R, but even in R 2 there are famous examples that challenge this intuition. Indeed, consider the set C R 2 (pictured right) given by C = {(0, y) : y [ 1, 1]} {( x, sin ( 1 x)) : x (0, ) }. This is called the Topolgist s Sine Curve, and while it is defined as the disjoint union of two sets, it is actually connected. Indeed, any open set containing the vertical strip {(0, y)} y [ 1,1] will necessarily contain Figure 2: Topologist s Sine Curve 7

8 some of the curve. Thus this set cannot be covered by two disjoint, open sets. (Actually, a stronger notion of connectedness is path-connectedness. Roughly speaking, a set is path-connected if one can draw a path between any two points in the set without leaving the set. At first glance, path-connectedness may seem equivalent to connectedness, but the topologist s sine curve is connected without being path-connected.) One important result involving connectedness helps us classify sets which are both open and closed. Proposition 24. Suppose that (X, τ) is a topological space. Then X is connected iff the only sets which are both open and closed in X are and X itself. Definition 25 (Compactness). Let (X, τ) be a topological space and let C X. We say that C is compact iff from any collection of open sets {U i } i I such that C i I U i, we can extract a finite collection of sets U i1,..., U in such that C n k=1 U i k. Such a collection {U i } i I in the definition of compactness is called an open cover of C. Thus, in words, a set C is compact if every open cover of C admits a finite subcover. Example 26. Compactness is somehow a generalization of closedness and boundedness. Indeed, we give examples here of a bounded set which is not closed and a closed set which is not bounded and prove that neither are compact. Consider R with the standard topology and consider the bounded set (0, 1). This set is not closed, and we show that it is not compact. Consider U k = (0, 1 1/k) for k N. We see that (0, 1) k=1 U k so {U k } forms an open cover of (0, 1). However, if we take any finite subcollection U k1,..., U kn of {U k } and let K = max{k 1,..., k N }. Then (0, 1) N n=1u kn = (0, 1 1/K). Thus there cannot be a finite subcover for this particular cover, and so this cover violates the definition of compactness and we conclude that (0, 1) is not compact. Similarly, consider the set [0, ). This set is closed, but is not bounded, and we show that it is not compact. Consider the collection U k = ( 1, k) for k N. We see [0, ) k=1 U k, but again, any finite collection U k1,..., U kn will satisfy [0, ) N n=1u kn = ( 1, K) where K = max{k 1,..., k N }, and so this open cover admits no finite subcover, and hence we conclude that (0, ) is not compact. By contrast, a closed and bounded subset of R like [0, 1] is compact, though this is not at all trivial to prove. In general topological spaces, it is easier to show that a set isn t compact, since this only requires exhibiting one example of an infinite cover that does not admit a finite subcover. In R (and more generally, in metric topologies), there are nice theorems which give concrete lists of properties which are equivalent to compactness. We cover some of these theorems in the Real Analysis portion of these notes. Proposition 27. Suppose that (X, τ) and (Y, σ) are topological spaces and that f : X Y is continuous. Let U X and consider the image of U under f defined by f(u).= {f(x) : x U} Y. If U is connected in X, then f(u) is connected in Y. Likewise, if U is compact, then f(u) is compact in Y. This proposition tells us that these properties of connectedness and compactness are in- 8

9 variant under continuous maps. Statements like this help us characterize topological spaces. Note however that the converse is not true: a continuous map could still map a disconnected set to a connected set (for example, the continuous function f(x) = x 2 on R maps the disconnected set ( 1, 0) (0, 1) to the connected set (0, 1)), or a non-compact set to a compact set (for example, the continuous map f(x) = sin(x) maps the non-compact set (0, ) to the compact set [ 1, 1]). If we want a continuous map to not change the structure of a space at all, we need to require something more. Definition 28 (Homeomorphism). Suppose that (X, τ) and (Y, σ) are topological spaces. A function f : X Y is called a homeomorphism iff the following four properties hold: 1. f is one-to-one, 2. f is onto, 3. f is continuous, 4. f 1 is continuous. If such a function f exists, the topological spaces (X, τ) and (Y, σ) are called homeomorphic and we write X = Y. In words, a homeomorphism between two topological spaces is a bicontinuous bijection; it maps each space bijectively and continuously to the other. Homeomorphic spaces share essentially all important properties in common, so when two spaces are homeomorphic we think of them as morally the same space. For this reason it can be important to identify whether two spaces are homemorphic. We make a few final statements about properties that homeomorphic spaces share, and give one example in conclusion. Proposition 29. Let (X, τ) and (Y, σ) be topological space, let U X and let f : X Y be a homeomorphism. Then U is open in X iff f(u) is open in Y, U is closed in X iff f(u) is closed Y, U is connected in X iff f(u) is connected in Y, U is compact in X iff f(u) is compact in Y, X is Hausdorff iff Y is Hausdorff. Example 30. Consider R with the usual topology. Any open interval (a, b) is homeomorphic to the interval (0, 1) under the map f : (a, b) (0, 1) defined by f(x) = x a b a, x (a, b). 9

10 Likewise, (0, 1) is homeomorphic to R under the map g : (0, 1) R defined by g(x) = tan ( π ( x 1 2)), x (0, 1). Homemorphism is an equivalence relation (if two spaces are homemorphic to the same space, they are homemorphic to each other); indeed we can always compose homemorphic maps and retain a homemorphism. Thus any interval (a, b) is homemorphic to R. 10

11 Real Analysis Real analysis is concerned with the rigorous underpinnings of calculus. However, when we teach calculus, we do everything formally and so everything is assumed to be nice (all the commong functions from calculus are smooth, for example). Now we make no such assumptions: analysis is largely about tearing down out intuition from calculus and building it back up again but with rigor. Accordingly, most real analysis courses start with the basic construction of R; for brevity, we leave this out and simply start talking about properties of sequences, functions, sets, etc. Much of this will overlap with the preceding topology notes, but often the same concepts are tackled in very different ways. Some of this will also be repeated from the Calculus I & II notes. We ll start by discussing general metric spaces, giving several definitions and theorems, and later specialize the conversation to R. Definition 31 (Metric Space). Let X be a set and let d : X X [0, ). We call d a metric on X (and call (X, d) a metric space) if the following three properties hold: 1. d(x, x) = 0 for all x X and d(x, y) > 0 when x, y X, x y, 2. d(x, y) = d(y, x) for all x, y X, 3. d(x, z) d(x, y) + d(y, z) for all x, y, z X. Metrics generalize the notion of distance to non-euclidean spaces. Example 32. The prototypical example of a metric space is R with the metric d(x, y) = x y. This can be generalized to R n. Indeed, in R n, we define the metric ( n ) 1/2 d(x, y) = x y.= (x i y i ) 2, x, y R n. i=1 Another example: for any set X, we can define the discrete metric d(x, y) = 0 if x = y and d(x, y) = 1 if x y. Definition 33 (Metric Topology). Let (X, d) be a metric space, x X and r > 0. Define the ball centered at x of radius r by B r (x) = {y X : d(x, y) < r}. Let τ be the topology generated by the set β = {B r (x) : x X, r > 0}. This is called the metric topology on X. In this topology, a set U X is open iff for all x U, there is r > 0 such that B r (x) U. We will also refer to τ as the topology generated by the metric d. Conversely, if we have a topological space (X, τ) and there is a metric d on X that generates τ, then we call τ metrizable. 11

12 For some example, the discrete metric on a set will generate the discrete topology. The standard metric d(x, y) = x y on R will generate the standard topology on R. Metric topologies have very nice structure. Most of the topological properties discussed above can be given new definitions only using the metric structure that d lends to X. Thus, the definitions in real analysis and topology may look different at a first glance, but they are always compatible. We discuss some properties of metric topologies now. Proposition 34. Let (X, d) be a metric space. The metric topology on (X, d) is Hausdorff. This is very easy to prove: if x, y X and x y then d(x, y) > 0. Then B ε (x) and B ε (y), where ε = d(x, y)/3, are open neighborhoods of x and y respectively which are disjoint, proving that the space is Hausdorff. Definition 35 (Limit of a Sequence). Let (X, d) be a metric space and let {x n } n=1 be a sequence in X. We say that x X is the limit of x n iff for all ε > 0, there is N N such that d(x, x n ) < ε for all n N. In this case, we say that x n converges to x and we write x n x or lim n x n = x. This definition of the limit is exactly as in calculus but generalized to arbitrary metric spaces. Limits give us a way to characterize closed sets in metric topologies. Definition 36 (Limit Points). Let (X, d) be a metric space and let U X. A point x X is called a limit point of U if there is a sequence {x n } in U such that x n x for all n N and x n x. Proposition 37. Let (X, d) be a metric space and let C X. Then C is closed in the metric topology iff for all sqeuences {x n } in C converging to a limit x X, we have that x C. In the terminology of the above definition, a subset of a metric space is closed iff it contains all of its limit points. Recall, in topology a set is closed iff the complement of the set is open. This theorem gives an alternate definition, and it is often times easier to check that a set contains its limit points than to check that its complement is open. Definition 38 (Closure). Let U X and let L = {x X \ U : x is a limit point of U.}. Then the closure of U is defined by U = U L. That is, the closure of U is the set U plus all of the limit points of U. Note, we already defined the closure in topology to be smallest closed set containing a set. Again, these notions are compatible: U defined in the definition above is the smallest set which is closed in the metric topology and contains U. In many metric spaces, we can build any point in the space by considering a smaller set and taking limits. Definition 39 (Dense Set). Let (X, d) be a metric space and let D X. We say that D is dense in X iff for all x X, there is a sequence {x n } in D such that x n x. Equivalently, 12

13 D is dense in X iff for all x X and ε > 0, there is y D such that d(x, y) < ε. Again, equivalently, D is dense in X if D = X. One last equivalent statement: D is dense in X if every open subset of X contains a least one point in D. Intuitively, a dense set is tightly packed into X; it may not include all elements, but the gaps between elements are infinitesimally small. In this way it seems like a dense set must contain most of the space, but this is a place where intuition fails. Indeed, a dense set can actually be quite small in a few different senses. We define one sense here and discuss it more when we discuss R later. Definition 40 (Countable Set). A set C is countable if there is a injective map f : C N. Equivalently, a set C is countable if the elements of C can be listed in a sequence: C = {c n } n=1. A set is C countably infinite if there is a bijective map f : C N. Definition 41 (Separable Space). Let (X, d) be a metric space (or more generally a topological space). We say that X is separable if there is a countable set D X which is dense in X: D = X. Again, intuitively, it may seem like a separable space needs to be small because it has a small dense set, but this intuition is not true in any meaningful sense. There are highly non-trivial separable spaces. Definition 42 (Cauchy Sequence). Let (X, d) be a metric space and let {x n } be a sequence in X. We call {x n } a Cauchy sequence iff for all ε > 0, there is N N such that d(x n, x m ) < ε for all n, m N. A Cauchy sequence is one that eventually begins to cluster together. Intuitively, we may think that if the sequence clusters together, it must cluster around some point and thus it will converge to that point. However, if the space X is missing some points, then the sequence may cluster around a missing point and thus fail to converge to any member of X. Thus we use Cauchy sequences to define a notion of a space not having any missing points. Definition 43 (Complete Space). We call a metric space (X, d) complete iff for all Cauchy sequences {x n } in X, there is x X such that x n x. Complete spaces are nice because to prove a sequence {x n } has a limit one first needs to identify a candidate x and then prove that d(x, x n ) becomes small. However, the candidate x may be difficult or impossible to identify. However, if the space is complete, to prove that a sequence converges, one no longer needs to identify a candidate; rather can instead prove that {x n } is a Cauchy sequence and conclude that it converges in that manner. Besides sequences, much of calculus is concerned with functions and their properties like continuity, differentiability and integrability. We can discuss continuity in general metric spaces; the other concepts require some of the structure of R, so we leave them for later. Definition 44 (Continuity). Let (X, d X ) and (Y, d Y ) be two metric spaces, let f : X Y 13

14 and let x X. We say that f is continuous at x iff for all ε > 0, there is δ = δ(x, ε) > 0 such that for z X, d X (x, z) < δ = d Y (f(x), f(z)) < ε. We say that f is continuous on X (or merely, f is continuous) iff f is continuous at every point x X. Equivalently, a function f is continuous on X iff for all x, y X and all ε > 0, there is δ = δ(x, y, ε) > 0 such that d X (x, y) < δ = d Y (f(x), f(y)) < ε. This is the exact notion of continuity that we presented in calculus, but generalized to metric spaces. Again, it is a useful exercise to prove that this notion of continuity is equivalent to the topological notion of continuity. Because metric spaces have nice structure, we can also characterize continuity in terms of limits of sequences. Theorem 45 (Sequential Criterion Theorem). Let (X, d X ) and (Y, d Y ) be metric spaces and let f : X Y. Then f is continuous iff for all sequences {x n } in X converging to a point x X, we have that f(x n ) f(x) in Y. A function satisfying the latter condition is said to be sequentially continuous, so this proposition tells us that a function between metric spaces is continuous iff it is sequentially continuous. In calculus, this would conclude our discussion of continuity. However, in analysis, we define more stringent definitions of continuity to more finely differentiate between classes of functions. Definition 46 (Uniform Continuity). Let (X, d X ) and (Y, d Y ) be two metric spaces and let f : X Y. We say that f is uniformly continuous iff for all ε > 0, there is δ = δ(ε) > 0 such that for all x, y X, d X (x, y) < δ = d Y (f(x), f(y)) < ε. At first glance this definition looks identical to the definition of continuity, but it is not. The subtle difference is in the order of the quantifiers. In the definition of continuity, the δ is allowed to depend on the particular x and y you are testing; in the definition of uniform continuity, δ cannot depend on x and y: there must be a uniform δ that only depends ε. In logical notation this different is expressed as such: f is continuous iff ( ε > 0)( x, y X)( δ > 0) ; d X (x, y) < δ = d Y (f(x), f(y)) < ε, whereas f is uniformly continuous iff ( ε > 0)( δ > 0)( x, y X) ; d X (x, y) < δ = d Y (f(x), f(y)) < ε. Thus uniform continuity is a stronger condition: if f is uniformly continuous, then f is continuous, but not vice versa. And even stronger notion of continuity is as follows. Definition 47 (Lipschitz Continuity). Let (X, d X ) and (Y, d Y ) be two metric spaces and let f : X Y. We say that f is Lipschitz continuous iff there is a constant L > 0 such that for all x, y X, d Y (f(x), f(y)) L d X (x, y). In this case, the smallest such L is called the Lipschitz constant of f. 14

15 Thus Lipschitz continuous function have an explicit bound on the distance between f(x) and f(y) in terms of the distance between x and y. If f is Lipschitz continuous with constant L, then for any ε > 0, we can take δ = ε/l and we will find that f satisfies the definition of uniform continuity; hence Lipschitz continuity is even stronger. With this we drop generality and talk specifically about the analytic and topological structure of R. Again, we will not explicitly construct the real numbers, but we ll present the rough idea, which is to start with rational numbers and define real numbers as limit points of Cauchy sequences of rationals. Definition 48 (Rational Numbers). Define the set of rational numbers to be those which can be written as a ratio of integers. That is, Q = { p : p, q Z, q 0}. With this q definition, the rational numbers form a field: they are an abelian group under addition and, neglecting the additive identity, they are an abelian group under multiplication. The rational numbers become a metric space with the metric d(a, b) = a b for a, b Q. Proposition 49. Q is countably infinite. That is, we can enumerate the rationals in a sequence Q = {q n } n=1. This requires what is called a diagonalization argument. One can list the rationals in a two-dimensional table where row k corresponds to the rationals whose denominator is k. Then one can traverse the table down each diagonal, assigning a natural number to each rational number. Proposition 50. Q is indiscrete in the sense that between any two rational numbers, one can find another rational number. Indeed, if a, b Q and a < b, then for large enough n N, we have a < a + 1 < b and a + 1 is still rational. n n From this proposition, it seems that the rational numbers do not have any large holes, and this may lead us to believe that the rational numbers are complete, but this is incorrect. Indeed, the rational numbers do not form a complete metric space. Proposition 51. There is no x Q such that x 2 = 2, but there is a Cauchy sequence {x n } in Q such that x 2 n 2. This proposition ruins completeness. Define f : Q Q by f(x) = x 2 for x Q. We see that f(0) = 0 and f(2) = 4, so intuitively because f cannot jump over points, we should be able to find x Q such that f(x) = 2, but we ve just asserted that this is impossible. Thus we conclude that either f is discontinuous because it jumps over a point (but it is easy to show that f is indeed continuous), or that Q is missing some points. It is the latter that is true. In the terminology of metric spaces, this shows that Q is not complete. However, for any metric space, we can define a notion of the completion of the space; that is, we can define a new metric space by adding some points, which is compatible with the original but is complete. This is how we define R. That is, the real numbers are defined as those numbers 15

16 which can be realized as limit points of Cauchy sequences of rational numbers. Thus any rational is real, but the numbers x such that x 2 = 2 are real without being rational. Though this is essentially the definition of R, we state this as a proposition. Proposition 52. Any real number is a limit of rational numbers. That is, Q is dense in R, and thus any open set in R contains rational numbers. Furthermore R is complete under the metric d(x, y) = x y, x, y R. Definition 53 (Irrational Numbers). The irrational numbers to be those which are real but not rational; that is, the irrational numbers are given by R \ Q = {x R : x Q}. Thus, for example, the solution to x 2 = 2 are irrational, and of course, since f : [0, ) [0, ) defined by f(x) = x 2 for x [0, ) is bijective, we can define an inverse map, and once we ve done, we denote the solutions to x 2 = 2 as 2 and 2. Other common irrational numbers are π and e. We asserted before that Q is a countable set. It is reasonable to ask if R is still countable since all numbers in R are simply limits of numbers in Q. Proposition 54. The set R of real numbers is not countable, and thus R\Q is not countable either. To prove this, one can use another type of diagonalization argument. If we assume that the numbers between 0 and 1 are countable, then we can list their decimal representations, but then it is not difficult to explicitly construct a number between 0 and 1 which is not accounted for in the list. In this sense Q is much smaller than R, but is still manages to be dense in R. While we know there are holes in Q, we might still wonder about the general structure of Q within R. Proposition 55. Q is disconnected in R. Indeed, we see that Q (, 2) ( 2, ). which shows that Q is contained in two disjoint open sets. We can make this even stronger. Proposition 56. The irrational numbers R \ Q are dense in R. Thus in between any two rationals, we can find an irrational, and we can use this to show that if a set of rational numbers has two points, then it cannot be connected; that is, Q is totally disconnected as a subset of R. This example that 2 Q displays something troubling about the structure of Q. Consider the set A = {x Q : x 2 < 2}. It is easy to see that this set is bounded (the elements of this set do not get arbitrarily large; for example, for x A, we will certainly have x < 5), but there is no tight upper bound in Q. That is, for any rational number q such that x < q for any x A, one could find a smaller rational number p < q such that x < p for all x A. In short: there is no least upper bound for this set in Q. This is a feature which is fixed by moving to R. 16

17 Definition 57 (Supremum & Infimum). Suppose that A R. The supremum of A is defined to be the least upper bound of A. That is, the supremum is the number S R (if such a number exists) such that x S for all x A, and if x M for all x A, then S M. Likewise, the infimum of the set A is defined to the be greatest lower bounded of A. That is, the infimum is the number I R (if such a number exists) such that x I for all x A, and if x m for all x A, then I m. When such numbers S, I exist, we write S = sup(a) and I = inf(a). Definition 58 (Bounded Set). that x M for all x A. A set A R is called bounded, if there is M > 0 such Proposition 59. Every non-empty bounded set in R has a finite supremum and infimum. (Indeed, if we allow the supemum or infimum to take the values ±, then every non-empty set in R has a supremum and infimum.) This is one more way in which we have filled in the holes when moving from Q to R. We want to further study the analytical and topological properties of R. We have already remarked that R is complete and thus every Cauchy sequence in R has a limit in R. As we said before, one advantage of this is that when testing if a sequence has a limit, we do not need to identify a candidate for the limit to prove convergence. We would like other such tests to characterize when limits in R exist. Theorem 60 (Monotone Convergence Theorem). Suppose that {x n } is a sequence in R which is increasing (that is, x n x n+1 for all n N) and bounded above (that is, there exists M > 0 such that x n M for all n N). Then {x n } converges (and in fact, {x n } will converge to the least M such that x n M for all n N). Likewise if {x n } is decreasing and bounded below then it converges. Loosely speaking, there are two possible ways for a sequence not to converge. It could either blow up to ± as with the sequence x n = n 2 or it could oscillate between certain values as with the sequence x n = ( 1) n. In the first case, no matter how we look at it, the sequence will always diverge, but in the latter case, if we look only along the even terms x 2n = ( 1) 2n = 1, then we have a stable sequence which converges. The succeeding definitions and theorems deals with this situation. Definition 61 (Subsequence). Suppose that {x n } n=1 is a sequence in R. A subsequence is a sequence {x nk } k=1 such that n k < n k+1 for all k N. Thus {x nk } {x n }. Theorem 62 (Convergence Along Subsequences). Suppose that {x n } is a sequence in R such that x n x R as n. Then every subsequence {x nk } will also satisfy x nk x as k. Theorem 63 (Bolzano-Weierstrass Theorem). Every bounded sequence has a convergent subsequence. That is, suppose that {x n } is sequence in R and there is M > 0 such 17

18 that x n M for all n N. Then there exists some susbsequence {x nk } which converges. Another way to state the above theorem is that if the sequence {x n } resides in the bounded set (a, b), then along a subsequence we can find a limit. If instead we consider the closed set [a, b], then this set contains its limit points and so the limit must also lie in [a, b]. With this is mind, we state a theorem characterizing compact sets in R. Theorem 64 (Heine-Borel Theorem). covers) iff it is closed and bounded. A subset of R is compact (in the sense of open Thus compact sets are precisely those which contain all their limit points and are not too large in either direction. With this in mind, the prototypical compact sets in R are the closed intervals [a, b] where a, b R, a < b. However, this theorem applies more generally in the metric topology on R n. In light of the Bolzano-Weierstrass Theorem, we can add another equivalent statement. Proposition 65. Suppose that C R. Then the following are equivalent: C is compact, C is closed and bounded, every sequence in C has subsequence converging to a point in C. Sets satisfying the third property are called sequentially compact and this theorem tells us that in R, sets are compact iff they are sequentially compact. Now a valid question is: why is compactness an important property? The definition of compactness is somewhat opaque, but compactness allows you to narrow your focus from infinitey many things to finitely many things. This especially comes in handy when dealing with functions. Indeed, we will move from here to discussing functions on R. Definition 66 (Bounded Function). Suppose that f : A R where A R. We say that f id bounded iff there is M > 0 such that f(x) M for all x A. That is, f is bounded if the image {f(x) : x A} is a bounded set in R. Proposition 67. Continuous functions from compact sets into R are bounded, and achieve maximum and minimum values. Specifically, suppose that C R, C is compact and f : C R is continuous. Then there are x min, x max C such that f(x min ) f(x) f(x max ) for all x C. [Note: this is not only asserting that f remains trapped between two extreme values, it is also asserting the existence of x min and x max where f meets those extreme values.] How does compactness come into play here? Consider, if f is continuous then the sets U n = {x C : f(x) < n} for n N are open since they are the pre-image of the open sets (, n). Also since f(x) R for all x C, for each x C we can find n N such that x U n. This shows that {U n } is an open cover of C. If C is compact, there is a finite subcover U n1,, U nk. However, these sets are nested, so this shows that C U N 18

19 where N = max{n 1,..., n K }. Then for all x C, we have f(x) < N, which shows that f is bounded from above. What has happened here? We began with infinitely many different bounds f(x) < n for n N each of which may have applied at different portions of C. Using compactness we were able to pare this down to a finite number of bounds, and then simply pick the largest one. Making a similar argument, we can get a lower bound, and thus the range {f(x) : x C} is bounded. Since the set is bounded, it has an infimum and supremum. The theorem also asserts that f will meet these values. How do we find the point that meets the supremum? We can contruct a maximizing sequence {x n } such that f(x n ) sup f(x) = sup{f(x) : x C}. x C By sequential compactness, the sequence has a subsequence {x nk } with a limit x max C and by continuity, we will have f(x nk ) f(x max ) and f(x max ) = sup x C f(x). Thus while compactness helps us arrive at the bound on f, sequential compactness helps us actually find the point where f achieves the bound. Above we defined not only continuity but also uniform continuity and Lipschitz continuity. We would like easy ways to identify which functions satisfy these stronger properties and this is somewhere where compactness can help a bit as well. Proposition 68. Continuous functions on compact sets are uniformly continuous. That is, suppose that C R and f : C R is continuous. If C is compact, then f is uniformly continuous. Again, we should observe how compactness comes into play. Fix ε > 0. Recall, if f is continuous at each point x C, then for each individual point, we can find δ x > 0 such that for y C, x y < δ x = f(x) f(y) < ε. Here we have (possibly) infinitely many different δ x > 0, but if we want to satisfy the definition of uniform continuity, we need to have a single δ > 0. If the number δ x > 0 works in the definition of continuity at x C, then any smaller number 0 < δ < δ x will also work. Thus one idea is to take the minimum over all such δ x > 0, and this minimal δ will work for all x C. However, the set {δ x } x C may not have a minimum and its infimum maybe zero, so this doesn t quite work. But we note that the sets (x δ x, x + δ x ) form any open cover of C. If C is compact, we can extrace a finite subcover (x 1 δ x1, x 1 + δ x1 ),..., (x K δ xk, x K + δ xk ) which still covers all of C. Now there are only finitely many δ xk to choose from; choosing the minimum δ = min{δ x1,..., δ xk } will provide a δ > 0 which works uniformly over all x C, proving that f is uniformly continuous. Again, we started with an infinite collection, and compactness allowed us to pare it down to a finite collection. Lipschitz continuity can also be easier to identify via compactness but in a slightly more complicated way. First, recall a few definitions and theorems from calculus (for more exposition regarding the calculus topic, one can look back to the calculus notes). Definition 69 (Lipschitz Continuity). Suppose that f : R R. We call f Lipschitz continuous if there is a constant L > 0 such that for all x, y R, f(x) f(y) L x y. 19

20 Definition 70 (Differentiability). is differentiable at x if the limit Suppose that f : R R and x R. We say that f f(x + h) f(x) lim h 0 h exists. In this case we call the limit f (x). We say that f is differentiable in a set A R if f is differentiable at all x A. Theorem 71 (Mean Value Theorem). Suppose that f : R R is differentiable in R. Then for any a, b R, a < b, there is c (a, b) such that f(b) f(a) b a = f (c). This, in turn, implies that f(b) f(a) = f (c) b a. Note the similarity between the last statement and the definition of Lipschitz continuity. There s a very formal similarity that hints at some connection of the form f (c) L. Indeed, we can make this precise. Proposition 72. Suppose that f : R R is differentiable on R. If the derivative f : R R is continuous, then f is Lipschitz continuous on any compact subset of R. If the derivative f is bounded, then f is Lipschitz continuous on all of R. If f is unbounded on some subset of R, then f is not Lipschitz continuous on that subset. This gives a rough equivalence between Lipschitz continuous functions and continuously differentiable functions. However, based on this continuous differentiability still seems a bit stronger than Lipschitz continuity and indeed, it is. Take for example, the function f(x) = x for x R. This function if Lipschitz continuous with Lipschitz constant 1 because of the reverse triangle inequality: f(x) f(y) = x y x y. However, this function is not differentiable on all of R. [In fact, a famous theorem states thata function is Lipschitz continuous functions iff it is differentiable almost everywhere 2 and the a.e. derivative is essentially bounded.] Finally, we discuss sequences of functions and the interplay between sequences of functions and the Riemann integral. Indeed, one of the reasons that the Lebesgue integral and the field of measure theory came about was because the Riemann integral does not play nice with sequences of functions, as we will see shortly. Definition 73 (Pointwise Convergence). Let A R and let {f n } be a sequence of functions f n : A R. We say that the sequence {f n } converges pointwise to a function 2 Here almost everywhere has a technical meaning. 20

21 f : A R iff for all x A, the sequence {f n (x)} in R converges to f(x). That is, {f n } converges pointwise to f iff for every x A and ε > 0, there is N = N(x, ε) N such that f n (x) f(x) < ε for all n N. Example 74. Consider the sequences f n, g n : [ 1, 1] R given by f n (x) = x n, and g n (x) = x for x [ 1, 1]. n Note that f n (0) = 0, f n (±1) = 1, for all n N. If x [ 1, 1] \ { 1, 0, 1}, then log x < 0 and so lim f n(x) = lim x n = lim e n log x = 0. n n n Thus f n converges pointwise to the function f : [ 1, 1] R defined by { 1, x = 1, 1, f(x) = 0, x ( 1, 1). Next for g n, note that for all x [ 1, 1], x = x 2 x = g n n(x). Conversely, if a, b 0 then a 2 + b 2 a + b (one can easily verify this inequality by squaring both sides), and so g n (x) = x n x + 1. n Combining the inequalities and subtracting x, we see Thus for all x [ 1, 1], we have 0 g n (x) x 1 n, x [ 1, 1]. lim g n(x) = x n and so g n converges pointwise to the absolute value function on [ 1, 1]. There are two interesting differences to point out between these examples. In the first example, we had a sequence of continuous functions which converged pointwise to a discontinuous function, which is somewhat disconcerting (this is similar to before when we had a sequence of rationals converging to an irrational; morally, this indicates that continiuous functions are incomplete with respect to pointwise limits ). The other difference is that proving the convergence of f n required special consideration for different values of x, whereas proving the convergence for g n did not. To address both of these, we introduce a stronger notion of convergence. 21