Mathematical Preliminaries

Transcription

1 Part I Mathematical Preliminaries Notation N Z R - subset (or set inclusion) - strict subset (or strict set inclusion) - superset - strict superset - element of - contains - the set of natural numbers - the set of integers - the set of real numbers 2 Sets, Functions, and Proofs 2.1 Sets and Functions A set is a collection of objects. We will work with this intuitive definition of a set without limiting what a set may contain 3. A set that contains no elements is called the empty set and is denoted by or {}. The union of two sets A and B, denoted A B, is a set that contains all elements that are contained in A and all elements that are contained in B, without repetition. The intersection of two sets A and B, denoted A B, is a set that contains only those elements that are contained in both A and in B. A set C is a subset of another set A if every element in C is also an element in A. The universal set U is the set containing all elements of every possible set. Note well that the universal set can be different depending on the context, and that the universal universal set is an ill-defined concept that leads to paradoxes (see previous footnote). Given two sets A and B, we can define the set difference A B or A\B to be the set of all elements in A that are not in B. Then, A = (A B) (A B). The complement of a set A, denoted Ā, is the set of all elements not in A, that is, the set of all elements that are in the universal set but not in A. Then, Ā = U A. The power set 3 This is naïve set theory, and suffers from the (Bertrand) Russell paradox: consider the set of all sets that do not contain themselves. Now does this set contain itself? 4

2 of a set A, denoted P (A), is the collection of all subsets of A. Notice that if the cardinality (see below for definition) of the set A is finite (and equal to a), then the number of subsets of A, i.e. the cardinality of the power set of A, is 2 a. Next, we (intuitively) define a map from one source set (the domain) to another target set (the codomain). Imagine the two sets written as a list, with one list written above the other. Now, imagine that there are some element-to-element links between the two sets but not within a set, where links are unique. This is the intuitive definition of a map. There are four types of maps, which describe the nature of the links between the two sets. A map is one-to-one if every element in the domain has no more than one link. A map is manyto-one if the domain has the property that every element of that set has no more than one link while the codomain contains at least one element that has more than one link. A map is one-to-many if every element in the codomain has no more than one link, but there is at least one element in the domain that has more than one link. A map is many-to-many if both domain and codomain contain at least one element that has more than one link. Given some map f, with domain A, the range of f is the set of all element in the codomain that have a link to some element of the domain. Thus, the range is a subset of the codomain. A function is a map from one set, called the domain of the function, to another set, called the codomain of the function, with the restriction that each element in the domain be mapped to exactly one element in the codomain. A function can be one-to-one or manyto-one (but not many-to-many or one-to-many). The cardinality of a set is the mathematical equivalent of the size of a set. For finite sets (set containing a finite number of elements) the cardinality is the same as the number of elements. For non-finite sets, the notion of size has to be extended to the realm of infinite numbers. In order to do this, we need to have a formal way of counting the elements in a set. Consider pairing an element from the set of unknown cardinality with an element of the set of natural numbers (the counting numbers 1, 2, 3...) denoted by N, following the natural order of the counting numbers. For example, if I had a basket of apples and I wished to know how many I had, I could have written down in ascending order the natural numbers and placed one and only one apple on each number starting from 1. The number of apples (that is, the cardinality of the set of apples) would then be given by the last number upon which I placed an apple. This pairing of an element from one set to another is captured by the definition of a function. Finally, we can formalize the notion of counting, and thus of cardinality. For any set A, we can define a counting function as a one-to-one function with domain A and codomain N with the restriction that for any element of the natural numbers to which there is a link, every 5

3 lesser natural number has a link. Now, if the counting function is surjective, in addition to being injective, i.e. bijective, then we say that the set is countably infinite. If the counting function has a range with a least upper bound, then the set is finite and has cardinality equal to the least upper bound. The empty set is considered to have cardinality 0. If there is no way to construct a counting function, because no one-to-one function exists, then the set is uncountably infinite. A set is countable if it is finite or countably infinite. Otherwise it is uncountable. Suppose A and B are subsets of some set X. Denote as A B the set of all points in A that are not contained in B i.e. A B {a A : a B}. Definition 2.1. Suppose X is the universal set. If A X, the complement of A, denoted A c, is the set X A. 2.2 Proofs In mathematics, a statement is a sentence that is either true or false. A proof is a sound argument for the truth of a particular statement expressed in mathematical language. An implication is a statement of the form If A is true, then B is true, where A and B are statements. Here, A is the hypothesis and B is the conclusion. Implications are often written as If A, then B or A implies B. There is a frequently used symbolic notation for the implication: A = B. A proposition is a true statement of interest to be proved the proof would accept the truth of some number of statements (the premises) and logically and cogently argue for the truth of the proposition. A truth table is a useful method for determining the truth value of complex statements. A theorem is a proposition that is subjectively considered to be of great import or value. Sometimes, because of the length of an argument for a theorem, the proof is broken into stages, with each linking proposition being proved as a lemma. So, lemmata (plural of lemma) are propositions whose subjective import derives not necessarily from its statement but from its role as a stage in the overarching construction of a proof of a theorem. However, occasionally a lemma has importance independent of the theorem for which it was constructed. Lastly, corollaries are propositions that follow almost immediately from a theorem; the proof of such a statement is usually trivial, but the subjective value of the knowledge of its truth is not. A little reflection will reveal that to be able to employ mathematical logic fruitfully one needs to know the truth value of some statements. Logic describes the relationships between statements, and describes the rules by which the truth value of a statement can be ascertained given a particular set of premises. However, to ground a particular systematic 6

4 body of knowledge one needs axioms. An axiom is a statement whose truth value is accepted without formal proof. The defense for the choice of a particular axiom, and the consequences of acceptance or rejection of a particular statement as axiomatically true is the bread and butter of theoretical economics. 4 Axioms are the atoms of a particular knowledge system, just as certain mathematical concepts that are without formal definition (see for example the definition of a set above) are atoms of a mathematical system. By definition, every statement is either true or false. Then, we can define logical operators on statements, analogous to the arithmetic operators plus and multiply. The operator AND denoted is a binary operator such that A B is true if and only if A is true and B is true. The operator OR denoted is a binary operator such that A B is true if and only if at least one of A and B is true. The operator NOT denoted is a unary operator such that A is true if and only if A is false. Using the language of operators, we can see now that implies, denoted operator? =, is a binary logical operator. What is the truth table for this Now, consider an arbitrary implication A = B. We can define three operations that take an implication and produce another implication. The converse of A = B denoted CON V is B = A. The inverse of A = B denoted IN V is A = B. The contrapositive of A = B denoted CONT R is B = A. What is the converse of the inverse of an implication? What is the inverse of the contrapositive? What is the inverse of the inverse of an implication? Suppose we have to prove that the implication A = B is true. We could attempt a direct proof, where we would assume A holds true and produce a chain of implications ending with the desired outcome i.e. A = C = D =... = E = B. Alternatively we could attempt an indirect proof, which comes in two varieties. First, we could directly prove the contrapositive B = A, which is equivalent to A = B. Second, we could assume that A = B is false i.e. A B is true, and then show that this assumption leads to a contradiction of a previously proved (or assumed) statement, a technique known as reductio ad absurbdum or proof by contradiction. A statement that is true or false conditional on the value of one or more variables is a conditional statement. E.g. x 2 + 3y = 5. Most statements one encounters are conditional statements. It is important to note that a conditional statement has a determinate truth value, conditional on the values of each of the variables upon which it depends. Intimately connected with conditional statements and implications are quantifiers, which 4 John von Neumann is often credited with introducing the axiomatic method in economic theory (for example, expected utility theory), particularly due to his previous work on the foundations of logic and set theory in mathematics and on the foundations of quantum mechanics in physics. Kenneth Arrow was an early (and successful) proponent of this approach, made especially famous in his work on social choice theory. 7

5 delineate the scope or domain in which the truth of a conditional statement or implication holds. There are two types of quantifiers: existential and universal. The existential quantifier can be recognized by the use of words such as there exists or there is/are, and can be denoted by. When such a quantifier is present, the truth of the (sub-)statement to which it is attached is determined by the possibility of constructing or otherwise proving the existence of at least one object satisfying the conditions of the statement. E.g. (There exist x, y such that x 2 + 3y = 5) is a statement (and not a conditional one), and furthermore is a true statement, since x = 1, y = 4 allows for the truth of the conditional statement. 3 Therefore, the only way for a statement with a single, existential quantifier to be false is for there to be no object that satisfies the conditional statement. Notice that there is an hidden assumption in the previous example. I argued that the statement is true by constructing an example (proof by construction). However, I assumed that x, y R. This is neither allowed nor disallowed by the statement, which implies that the truth value of the statement is itself conditional on the domain of the variables x and y. Suppose the statement read There exist x, y such x 2 + 3y = 5 and x, y N. Then, the statement would be false, since there is no pair of values for the variables that would satisfy the expression and the domain restrictions. The universal quantifier can be recognized by the use of words such as for all/every/any and can be denoted by. N.B. For some is not a universal quantifier, but an existential quantifier, even though the word for appears. When a universal quantifier is present, the truth of the (sub-)statement to which it is attached is determined by the possibility of constructing or otherwise proving the existence of at least one object not satisfying the conditions of the statement. Any such object would prove the statement false. E.g. (For all x, y, x 2 + 3y = 5) is false since x = 1, y = 1 yields a conditionally false statement. Sometimes, theorems or other statements involve the negation of quantifiers. Any statement of the form ( x, A(x)), where A(x) is a conditional statement, can be written as x, such that A(x). Thus, when negated, the universal quantifier becomes an existential quantifier, with the attached conditional statement becoming negated. A similar algorithm allows for the negation of an existential quantifier: ( x, such that A(x)) is equivalent to x, A(x). One common method of proof that may be employable is the proof by induction. The fundamental principle behind induction is that if S N such that (S 1) (n S = (n + 1) S), then S = N. Thus, proof by induction can be used whenever the statement to be proved has a universal quantifier with domain N. The proof requires two steps. The first step is to show the truth of the conditional statement for some particular m N. The second step is to show that the conditional statement is true for some n N if it is true for 8

6 n 1 (or any m < n, m N). 2.3 Useful Facts Theorem 2.1 (DeMorgan s Laws). Let X be some set, and suppose V a X for every a A, where A is some index set. Then, 1. ( a A V ) c a a A V a c 2. ( a A V ) c a a A V a c Definition 2.2. Let f : X Y. 1. The function f is injective if given x, x X, f(x) = f(x ) implies x = x. 2. The function f is surjective if for all y Y, there exists an x X such that f(x) = y. 3. The function f is bijective if it is injective and surjective. 4. If A X, denote by f(a) the set of all images of points in A i.e. f(a) {y Y : f(a) = y, a A}; we call f(a) the image of A. 5. If B Y, denote by f 1 (B) the set of all points in X whose images are in B i.e. f 1 (B) {x X : f(x) B}; we call f 1 (B) the preimage of B. Do not be confused by the notation f 1 (B). In particular, it is not a function like an inverse function. However, if f is bijective, then the preimage notation can be interpreted as an inverse function. It is best to think of f 1 as an operator acting on subsets of the range space. Proposition 2.2. Let f : X Y, A, A X, and B, B Y. Also, let A be an arbitrary collection of subsets of X and B be an arbitrary collection of subsets of Y. Then 1. f 1 satisfies the following: (a) B B implies f 1 (B) f 1 (B ). (b) f 1 (B B ) = f 1 (B) f 1 (B ). (c) f 1 ( B B B) = B B f 1 (B). (d) f 1 ( B B B) = B B f 1 (B). 2. Also, f satisfies the following: (a) A A implies f(a) f(a ). 9

7 (b) f(a A ) f(a) f(a ); equality obtains if f is injective. (c) f( A A A) = A A f(a). (d) f( A A A) A A f(a); equality obtains if f is injective. 3. Finally, f and f 1 satisfy: (a) A f 1 (f(a)); equality holds if f is injective. (b) B f(f 1 (B)); equality holds if f is surjective. Proof. We will use an element argument to demonstrate that B B = f 1 (B) f 1 (B ). Suppose B B and let x f 1 (B) be some arbitrary element. Then f(x) B, by definition of f 1, and so f(x) B, since B B. Thus, x f 1 (B ), by definition of f 1. The rest of the proof is left as an exercise. See Exercise 2.1. Exercise 2.1. Prove all the items in Proposition 2.2. For those statements that don t hold with with equality (items 2b, 2d, 3a, 3b), provide examples to show why equality fails to hold. 3 Real Vector Spaces Let V be a nonempty set. Definition 3.1. A real vector space is a set V together with two binary operators (+ and.) that satisfy the follow axioms (where u, v, w V and α, β R): 1. Associativity of vector addition: u + (v + w) = (u + v) + w 2. Commutativity of vector addition: u + v = v + w 3. Identity element of vector addition: There exists an element o V, the zero vector, such that v + o = v 4. Inverse element of vector addition: For all v there exists an element w, the additive inverse, such that v + w = o 5. Distributivity of scalar multiplication over vector addition: α.(u + v) = (α.u) + (α.v) 6. Distributivity of scalar multiplication over field addition: (α + β).u = (α.u) + (β.u) 7. Consistency of scalar multiplication with field multiplication: α(β.u) = (αβ).u 10

8 8. Identity element of scalar multiplication: 1u = u where 1 R is the identity element of the field R Vector spaces are also called linear spaces. The most familiar real vector space is R, which is also a field. The Euclidean spaces R n for finite n are also frequently encountered real vector spaces. A perhaps less familiar real vector space is the space of all real-valued continuous functions on the interval [0, 1]. Exercise 3.1. Show that the C([0, 1]), the set of all real-valued continuous functions on the interval [0, 1] is a real vector space. Definition 3.2. A normed vector space is a vector space V together with a function ν : V R, called a norm, such that for all u, v V, α R: R n. 1. ν(u) 0 and ν(u) = 0 u = o 2. ν(α.u) = α ν(u) 3. ν(u + v) ν(u) + ν(v) The norm formalizes the notion of a length of a vector. Consider the Euclidean space The adjective Euclidean derives from the use of the Euclidean norm, which for a vector v R n is defined to be n i=1 v2 i, where the v i is the i-th component when written as a linear combination of the standard orthonormal basis vectors. Definition 3.3. An real inner product space is a real vector space V together with a function, : V V R, called the inner product or dot product, such that for all u, v, w V, α R: 1. Positive-definiteness: u, u 0 and u, u = 0 u = o 2. Symmetry: u, v = v, u 3. Linearity: αu, v = α u, v and u + v, w = u, w + v, w For ease of notation, I will generally denote u, v as u v. Inner products formalize the notion of angles between vectors. For our spaces of choice, the Euclidean spaces R n, for n N, the usual inner product is defined by u v n i=1 (u iv i ). Thus, ν(u) = u u. In fact, if, is an inner product, then ν( ), is a norm. 11

9 4 Metric Spaces Let X be a nonempty set. Definition 4.1. A function ρ : X X R is a metric if, for any x, y, z X, 1. Non-negativity and properness: ρ(x, y) 0 and ρ(x, y) = 0 x = y 2. Symmetry: ρ(x, y) = ρ(y, x) 3. Triangle Inequality: ρ(x, y) ρ(x, z) + ρ(z, y) A metric space is a nonempty set X together with a metric ρ, denoted (X, ρ). Metrics formalize the notion of distance between points or elements of a set, and metric spaces allow for notions of convergence and of continuity, as we shall seen soon, though they are not the most basic way to formalize these notions. The (finite-dimensional) Euclidean spaces R n are metric spaces (in fact, they are complete metric spaces, as we shall soon see), with the metric being derived from the Euclidean norm as follows: ρ(x, y) ν(x y). These spaces are the workhorse for our exploration of classical and nonlinear programming. Exercise 4.1. Suppose that (X, ν) is a normed vector space. Show that (X, ρ) is a metric space when x, y X, ρ(x, y) ν(x y). The same set could be associated with many different metrics. An example of a different metric for R n is the taxicab or Manhattan metric, which is defined for x, y in the set X by n i=1 x i y i. In general, the metric used with a particular set can alter the mathematical properties of the associated metric space, but for finite-dimensional Euclidean spaces, the choice of metric does not affect the continuity properties of functions on the space (for the set of metrics derived from p-norms). Definition 4.2. Two metrics ρ 1 and ρ 2 defined for some set X are strongly equivalent if there exist positive constants α, β R such that for all x, y X, αρ 1 (x, y) ρ 2 (x, y) βρ 1 (x, y) Exercise 4.2. Show that the standard Euclidean metric ρ(x, y) i (x i y i ) 2 is strongly equivalent to the taxicab metric. 12

10 5 Analysis and Topology of Metric Spaces 5.1 Open Sets and Topology Let (X, ρ) be a metric space. Definition 5.1. An ɛ-ball about x, denoted B ɛ (x; ρ), is the set {y X : ρ(x, y) < ɛ}, where ɛ is a positive real number. When understandable by context, the notation for the metric will be suppressed. Definition 5.2. A set U X is open if for all x U, there exists ɛ > 0 such that B ɛ (x) U. It should be clear that ɛ-balls are open sets. Definition 5.3. An open neighborhood of a point x X is an open set U X such that x U. Definition 5.4. A point x X is an interior point of a set A X if there exists ɛ > 0 such that B ɛ (x) A. The interior of a set A, denoted inta, is the set of all interior points of A. The point x is a boundary point of the set A if for all ɛ > 0, B ɛ (x) A and B ɛ (x) (X A). The boundary of a set A, denoted bda, is the set of all points x X that are boundary points of A. Exercise 5.1. Show that the the interior of a set A X is equal to the union of all open subsets of X that are also subsets of A. Definition 5.5. A set A X is closed if bda A. Note that a set could be neither open nor closed. Also, a set could be both open and closed. This is amusingly depicted in the following web comic: com/394. Remark 1. A set A is open if and only if every point is an interior point i.e. A = inta. Definition 5.6 (Closure). The closure of a set A X, denoted cla, is the intersection of all closed sets containing A. Note that the cla is a closed set. Definition 5.7 (Bounded Set). Let (X, ρ) be a metric space. A set Y X is bounded if there exists x Y and r R + such that Y B r (x, ρ). Definition 5.8 (Totally Bounded Set). Let (X, ρ) be a metric space. A set Y X is totally bounded if for all ɛ R +, there exists a finite subset Z Y such that Y z Z B ɛ(x, ρ) i.e. the set Y can be covered by finitely many ɛ-balls, for any ɛ > 0. 13

11 Exercise 5.2. Show that a totally bounded set must also be bounded. Demonstrate with an example that the converse is not true. Corollary 5.1. For the Euclidean spaces R n, every bounded set is totally bounded. Hence, the definitions are equivalent for these spaces. Definition 5.9 (Metric Topology). Given a metric space (X, ρ), the metric topology induced by ρ is the set τ of all open subsets of X, where the open sets are defined as in Definition 5.2. Exercise 5.3. Let (X, ρ) be a metric space with the metric topology τ. Show that 1. the sets X and are both open and closed 2. an arbitrary union of open sets is open 3. the finite intersection of open sets is open 4. the complement of an open set is closed (and vice-versa) The first three items in the exercise 5.3 could be taken as the axioms for an arbitrary set X together with a collection τ of subsets of X to define a topological space (X, τ). Therefore, while every metric space has an associated topology, one could study a space with a topology without a metric or with a topology other than the one induced by the metric. Definition 5.10 (Topological Space). Given an arbitrary set X, a topology τ on X is a collection of subsets of X that satisfies the following conditions: 1. X and are elements of τ 2. τ is closed under arbitrary unions i.e. for any subcollection τ τ, ( U τ U ) τ 3. τ is closed under finite intersections i.e. for any finite subcollection τ τ, ( U τ U ) τ Member of these sets are called open sets, and complements of open sets are called closed. Definition 5.11 (Topological Base). Let X be a space. Suppose B is a collection of subsets of X such that: 1. B B B = X 2. For every x B 1 B 2, B 1, B 2 B, there exists B 3 B such that x B 3 B 1 B 2. 14

12 Then B is a base (or basis) for the topology τ, and τ is generated by B as follows: a set U X is open (U τ) if for any x U, there exists B B such that x B U. Notice that if one has a topology τ then a collection B is a base if every open set is the union of base elements. Moreover, every union of base elements is open. Example 5.1. The intervals (a, b), a < b form a base for the standard topology on R. Definition 5.12 (Subspace Topology). Let (X, τ) be a topological space. For Y X, the subspace topology (or relative topology or induced topology) of Y is the collection τ Y {U Y : U τ} i.e. the restriction of open sets in X to the set Y. Definition 5.13 (Box Topology). Let (X a, τ a ), a A be a family of topological spaces, indexed by A. The box topology of the Cartesian product X a X a is the topology generated by the base B box { a U a : U a τ a } i.e. every open set in X is the union of sets formed by the cartesian of product of sets open in X a. Definition 5.14 (Product Topology). Let (X a, τ a ), a A be a family of topological spaces, indexed by A. The product topology of the Cartesian product X a X a is the topology generated by the base B product, every element of which is formed by the cartesian product of sets a A Y a, Y a = X a for all a A where A A is a finite set, and Y a τ a for all a A A. Thus, the base consists of the cartesian product of entire spaces except for a finite number of the indices, for which the entire space is replaced with some set open in that particular space. Notice that if the index set A in the above definitions is finite, then the two bases are the same, and thus the two topologies are equivalent. Generally, when considering cartesian products of topological spaces, we will assume unless otherwise stated that the topology of the product space is the product topology. Definition 5.15 (Projection Mapping). Let (X a, τ a ) be a family of topological spaces indexed by the set A. The projection mapping associated with index b A is the function π b : X X b, where X a A X a, such that π b ((x a ) a A ) = x b. Thus, the projection mapping of an index b associates a point in the cartesian product with its bth coordinate. Definition 5.16 (Bounded Metric). Let (X, ρ) be a metric space. The metric is bounded if there exists M such that ρ(x, y) M for all x, y X. Thus, we have a bounded metric if the metric space is itself a bounded set. A metric need not bounded, but given any metric space (X, ρ), we could construct a bounded metric ρ ρ 1+ρ [0, 1]. Notice that ρ preserves the ordering of distances between 15

13 points i.e. ρ(x, y) ρ(u, v) ρ (x, y) ρ (u, v). In fact, there are other bounded metrics one could define that preserve the ordering of ρ. Definition 5.17 (Ordinal Equivalence). Let (X, ρ) be a metric space. A metric ρ is ordinally equivalent to ρ if for all x, y, u, v X, ρ(x, y) ρ(u, v) ρ (x, y) ρ (u, v). Definition 5.18 (Equivalence of metrics). Two metrics ρ 1 and ρ 2 defined for some set X are (topologically) equivalent if they generate the same topology. In particular, the two metrics are equivalent if for all x X and any ɛ > 0 there exist ɛ > 0 and ɛ > 0 such that B ɛ (x; ρ 1 ) B ɛ (x; ρ 2 ) and B ɛ (x; ρ 2 ) B ɛ (x; ρ 1 ) Exercise 5.4. Suppose ρ is a metric on X. Show that for any strictly increasing, continuous, subadditive function f : R + R +, where f(0) = 0, the function ρ f ρ defines a metric that is equivalent to ρ; a function f is subadditive if for any x, y, f(x + y) f(x) + f(y). Don t forget to prove that ρ satisfies the conditions to be a metric. Conclude that the metric ρ ρ 1+ρ is equivalent to ρ. Exercise 5.5. Show that ordinally equivalent metrics are equivalent metrics. Exercise 5.6. Show that strong equivalence of two metrics for a space X implies the metrics are equivalent. Note that the converse is not true arbitrary metric spaces, because a bounded metric can be equivalent to an unbounded metric, but cannot be strongly equivalent to it, because strong equivalence preserves the boundedness property (can you see why?). Definition 5.19 (Limit Point). For some topological space (X, τ), a point x is a limit point of a set A X if every open neighborhood of x intersects A at some point other than x itself. Notice that a limit point of a set need not be in the set. Closed sets exhibit the property that they contain all their limit points, which is a corollary to the following exercise. Exercise 5.7. Show that closure of a set A X is the union of A with the set of limit points of A. Definition 5.20 (Denseness). Let (X, τ) be a topological space. A subset Y Z is dense in Z if the closure of Y contains Z. Definition 5.21 (Separable). A metric space (X, ρ) is separable if there is a dense subset Y X that is countable. Remark 2 (Density of Rationals). The space of real numbers R are separable, because the rational numbers are a countable set that is dense in R. Exercise 5.8. Prove Remark 2. 16

14 5.2 Sequences Definition A sequence in X is a function a from N to X. A sequence is usually denoted (x n ), where x n x(n), n N. Definition 5.23 (Convergence: Metric). Let (X, ρ) be a metric space. A sequence (x n ) in X converges to x if, for every ɛ > 0, there exists an N N such that whenever n N, ρ(x n, x) < ɛ. We say that such an x is the limit of the sequence, with the notation being lim x n = x. A sequence that does not converge is said to diverge. This definition of convergence will not work for a topological space without a metric. We can define convergence of a sequence more generally for such spaces. Definition 5.24 (Convergence: Topological). Let (X, τ) be a topological space. A sequence (x n ) in X converges to x if for every open set U x, the sequence is eventually contained in the set U i.e. there exists an N such that x n U whenever n N. Exercise 5.9. Show that the two definitions of convergence of a sequence (x n ) are equivalent for metric spaces. Definition 5.25 (Sequentially Closed). Let (X, τ) be a topological space. A set Y X is sequentially closed if for every convergent sequence (x n ) contained in Y (where convergence is relative to the topology of X) the limit of the sequence is in Y. Theorem 5.2. If (X, ρ) is a metric space, then a set is closed if and only if it is sequentially closed. For general topological spaces, it is only true that a closed set is sequentially closed. The converse does not hold for an arbitrary topological space. Definition 5.26 (Subsequence). For some space X, let (x n ) be a sequence in X and consider an increasing sequence of natural numbers (m i ). This increasing sequence (m i ) produces a unique subsequence, (a mi ), of the original sequence. Note that the generated subsequence is itself a sequence. Definition 5.27 (Cauchy Sequence). Let (X, ρ) be a metric space. A sequence (x n ) is a Cauchy sequence if, for all ɛ > 0, there exists N N such that for all m, n N, ρ(a n, a m ) < ɛ. Notice that Cauchy sequences can only be defined for metric spaces, and not for topological spaces in general. Thus, completeness is not a topological property, because two equivalent metrics could yield different completeness properties. 17

15 Definition 5.28 (Completeness). Let (X, ρ) be a metric space. A set Y X is complete if every Cauchy sequence in Y has a limit in Y. Remark 3. Our favorite space R n is a complete metric space. Definition 5.29 (Bounded Sequence). Let (X, ρ) be a metric space. A sequence (x n ) is a bounded sequence if the set {x n } is a bounded set. 1. Every convergent sequence is bounded 2. Let lim a n = a and lim b n = b, where (a n ) and (b n ) are sequences in R. Then, (a) lim ca n = ca, c R (b) lim(a n + b n ) = a + b (c) lim(a n b n ) = ab (d) lim(a n /b n ) = a/b, b 0 (e) (a n 0, n) = a 0 (f) (a n b n, n) = a b (g) ( c R, c b n, n) = c b. A similar statement with the inequalities reverse also holds. 3. Every monotone and bounded sequence converges. 4. Subsequences of a convergent sequence converge to the same limit as the original sequence. Exercise 5.10 (Closed Sets Inherit Completeness). Let (X, ρ) be a complete metric space. Show that any closed subset Y X is also complete. 5.3 Continuity Definition 5.30 (Continuity at a point: Topological definition). Let (X, τ X ) and (Y, τ Y ) be topological spaces. A function f : X Y is continuous at x, if for all open neighborhoods V of f(x), there exists an open neighborhood U of x such that f(u) V i.e. the pre-image of open neighborhoods of f(x) are open. Definition 5.31 (Continuity at a point: Cauchy-Weierstrass definition). Let (X, ρ X ) and (Y, ρ Y ) be metric spaces. A function f : X Y is continuous at x, if for all ɛ > 0, there exists δ > 0 such that for all x X, ρ X (x, x ) < δ implies ρ(f(x), f(x )) < ɛ. 18

16 Definition 5.32 (Sequential Continuity at a point: Heine definition). Let (X, τ X ) and (Y, τ Y ) be topological spaces. A function f : X Y is sequentially continuous at x, if for all sequences (x n ) in X that converge to x, the sequence (f(x n )) converges to f(x) i.e. sequentially continuous functions preserve limits. Definition A function f : X Y is continuous if it is continuous at every point x X. A function f : X Y is sequentially continuous if it is sequentially continuous at every point x X. For metric spaces, all both definitions of continuity 5.30 and 5.31 are equivalent. Moreover, for metric spaces continuity and sequentially continuity are equivalent. However, in more general topological spaces, sequential continuity does not imply continuity, but the converse is still true. Exercise For arbitrary topological spaces X and Y, show that any continuous function f : X Y is sequentially continuous. Exercise Suppose f : R n R is continuous under the Euclidean metric. Show that function f is continuous under any metric on R n that is equivalent to the Euclidean metric. Exercise Show that the composition of two continuous functions is continuous. Definition 5.34 (Uniform Continuity). A function f : X R is uniformly continuous if for all ɛ > 0 there exists a δ > 0 such that for any x, y X if ρ(x, y) < δ, then f(x) f(y) < ɛ. Notice the slight change in the order of the quantifiers in the definition of uniform continuity from the Cauchy-Weierstrass definition of continuity. Exercise Suppose f : R n R is uniformly continuous under the Euclidean metric. Show that function f is uniformly continuous under any metric on R n that is strongly equivalent to the Euclidean metric. Proposition 5.3 (Continuity of Projection Mappings). Let (X a, τ a ) be a family of topological spaces indexed by the set A, and define X a X a. The projection mappings are continuous in both the product and box topologies. Proof. Suppose U b X b is an open set. Then the preimage of U b under the projection mapping π b is the set a Y a where Y a = X a for all a b and Y b = U b. But this set is an element of both the base B box and B product and is therefore open in both the box and the product topology. Thus, the preimage of open sets are open for any projection mapping, and so these mappings are continuous. 19

17 Theorem 5.4. Let (X a, τ a ) be a family of topological spaces indexed by the set A, and define X a X a with the product topology. Suppose f : Y X is defined by f(y) (f a (y)) a A, where f a : Y X a for every a. Then f is continuous if and only if f a is continuous for all a A. The previous theorem is not true for infinite cartesian products with the box topology. The following provides a simple counterexample. Example 5.2. Suppose f : R R. Suppose that f n : R R is defined by f n (t) = t for all n N. Thus, f(t) = (t, t, t,...). For each n, f n is continuous is the standard topology of R. However, f is not continuous when R has the box topology. Consider the set U = ( 1, 1) ( 1 2, 1 2 ) ( 1 3, 1 3 ). It is clear that U is open in R under the box topology, since ( 1 n, 1 n ) is open in R for any n N. However, f 1 (U) is not open in R. To demonstrate this, suppose to the contrary f 1 (U) were open. Then it would have to be an interval around 0, say ( ɛ, ɛ), which implies that f(( ɛ, ɛ)) U. Applying the projection mapping to the left side of the previous inclusion yields π n (f(( ɛ, ɛ))) = f n (( ɛ, ɛ)) = ( ɛ, ɛ) and to the right side yields π n (U) = ( 1, 1 ). Thus, ( ɛ, ɛ) ( 1, 1 ) for all n, which yields n n n n a contradiction since ɛ-interval could satisfy this, and since the set {0} is not open. 5.4 Compactness The most general definition, one that works for an arbitrary topological space, involves the notion of covers. Definition 5.35 (Open Cover). Let (X, τ) be a topological space, and F = {U α τ : α A} be an indexed family of open sets, where A is an index set. Then, F is an open cover of X if X α A U α. Definition 5.36 (Finite Subcover). Given an open cover F of X, a finite subcover is a finite subcollection of set from the original open cover F whose union still contains X. Definition 5.37 (Compact Set: Heine-Borel (Topological) definition). A set S X is compact if every open cover of S has a finite subcover. The topological definition is quite abstract and at this stage obscure; I include it for the sake of completeness of exposition. A somewhat more useful but still abstract definition involves the finite intersection property. Definition 5.38 (Finite Intersection Property). A collection of sets A has the finite intersection property if every finite subcollection {A 1,..., A m } has a nonempty intersection i.e. m i=1 A i. 20

18 Theorem 5.5. A set S X is compact if and only if every collection of closed subsets of S, A, with the finite intersection property has a nonempty intersection i.e. A A A. For metric spaces, the following notion, sequential compactness is equivalent to compactness, and is for us a more useful definition. Definition 5.39 (Sequential Compactness). For a topological space (X, τ), a set S X is sequentially compact if every sequence in S has a subsequence that converges to a limit that is also in S. If X is a metric space, then sequential compactness is equivalent to compactness. Theorem 5.6 (Heine-Borel Theorem). A nonempty set S R n (with the Euclidean metric) is compact if and only it is closed and bounded. The Heine-Borel Theorem allows us an easy characterization of compact sets in R n. Compact sets are useful because they behave as though they are finite sets (hence the word compact). In economics, we often assume that the sets we are working with are compact, particularly because of the Weierstrass Theorem. The theorem makes it easy to identify whether a given set from a Euclidean space is compact or not. There is a generalization of this theorem to metric spaces that requires some strengthening of the conditions. Theorem 5.7. Let (X, ρ) be a metric space. A set Y X is compact if and only if it is complete and totally bounded. Theorem 5.8 (Bolzano-Weierstrass Theorem). Every bounded sequence (x n ) in R n has a convergent subsequence. Equivalently, a subset of R n is sequentially compact (hence compact) if and only if it is closed and bounded. For metric spaces, the Bolzano-Weierstrass Theorem is essentially the same as the Heine- Borel Theorem because of the equivalence of the compactness and sequential compactness. The following is a crucial theorem from which the Weierstrass Extreme Value Theorem follows quite simply. Theorem 5.9 (Continuous Mappings Preserve Compactness). Let (X, τ X ) and (Y, τ Y ) be topological spaces, and suppose f : X Y is a continuous function. Then for any K X that is compact, f(k) is compact. Proof. Let A Y {V a : a A} be an open covering of the image f(k) of a compact set K X. Now, since f is continuous, the pre-image of every member of the collection A Y is open in X, so we have an open covering A X {f 1 (V ) : V A Y } of K. Since K is compact, every open cover has a finite subcover, and so there exists a finite subset A A 21

19 such that K a A,V a A Y f 1 (V a ). Now, since A defines a subcover of K relative to the collection A X, it defines a subcover of f(k) relative to the collection A Y. But A is finite and so we have a finite subcover for f(k), proving compactness of f(k). Theorem 5.10 (Uniform Continuity Theorem). Let (X, ρ X ) and (Y, ρ Y ) be metric spaces. If f : X Y is a continuous function, and X is a compact space, then f is uniformly continuous. The following theorem is a very useful result that says the Cartesian product of compact spaces is compact. Theorem 5.11 (Tychonoff Theorem). Suppose (X a, τ a ) is a compact space for any a A, where A is some index set. Then a A X a is compact in the product topology. Proposition The following are some useful results about bounded sets and compact sets: 1. The union of an arbitrary collection of bounded sets is not necessarily bounded. 2. The union of a finite collection of bounded sets is bounded. 3. The intersection of an arbitrary collection of bounded sets is bounded. 4. The sum of two bounded sets is bounded. 5. The union of an arbitrary collection of compact sets is not necessarily compact. 6. The union of a finite collection of compact sets is compact. 7. The sum of two compact sets is compact. 8. Closed subsets of compact spaces are compact. Exercise Prove item 8 in Proposition Connectedness Definition 5.40 (Connectedness). A space (X, τ) is connected if there do not exist two nonempty open disjoint sets U and V such that X = U V. A subset S X is connected in X if it is a connected space under the subspace topology. Proposition A space (X, τ) is connected if and only if the only sets that are both open and closed are X and. 22

20 Proof. Assumed connectedness of X. Suppose U X is nonempty and open. Then, U c is closed. But connectedness implies that U c is no open. Since the complement of every closed set is an open set, and since U is a generic nonempty open strict subset of X, generically every nonempty closed strict subset of X is not open. The proof of the other direction is trivial. Proposition 5.14 (Results about Connected Sets). Some results involving connected sets: 1. Continuous maps preserve connectedness i.e. the image of a connected set under a continuous function is connected. 2. Finite Cartesian products of connected sets are connected. Arbitrary Cartesian products of connected sets are connected under the product topology, but not the box topology. 3. The real line R is connected, as are intervals and rays (intervals that are unbounded on one side). As you continue your study of economics, you will find fixed point theorems pop up everywhere in microeconomics, because of their usefulness in proving equilibrium existence, from Walrasian equilibrium in the Arrow-Debreu-McKenzie-Nikaido general equilibrium model to Nash equilibrium in game theory. Fixed point theorems generally state that for some mapping ψ : Y Y, for some space Y, there exists a solution to the equation ψ(y) = y. You may not realize this, but you are probably already familiar with a fixed theorem, just not by that name. Acemoglu argues, convincingly, that the Intermediate Value Theorem has the quality of a fixed point theorem. Let us first see a statement of the theorem. Theorem 5.15 (Intermediate Value Theorem). Let (X, τ) be a connected topological space. Suppose f : X Y is a continuous function, where Y R endowed with the standard (subspace) topology 5. If a, b X and there exists z R such that f(a) z f(b), then there exists c X such that f(c) = z. To see why the Intermediate Value Theorem resembles a fixed point theorem consider a function f : X X, where X is a compact, connected subset of R i.e. X = [a, b], a b. Then, if f is continuous there exists c [a, b] such that f(c) = c. This follows quite simply from an application of the Intermediate Value Theorem. 5 The theorem could be generalized by taking the space Y to be any ordered space endowed with the order topology. 23

21 5.6 Sequences of Functions Suppose (f n ) is a sequence of functions from some set X to a metric space (Y, ρ). Definition 5.41 (Pointwise Convergence). The sequence (f n ) converges pointwise to a function f if for all x X, the sequence (f n (x)) converges to f(x). In notation, for all x X, for all ɛ > 0 there exists N such that for all n N, ρ(f n (x), f(x)) < ɛ. Definition 5.42 (Uniform Convergence). The sequence (f n ) converges uniformly to a function f if for all ɛ > 0 there exists N such that for any x X and for all n N, ρ(f n (x), f(x)) < ɛ. Equivalently, we have uniform convergence if lim sup{ρ(f n (x), f(x)) : x X} = 0. Notice the change in the order of quantifiers that is similar to the swap for uniform continuity. Pointwise convergence looks at the convergence of the function at a point, treating each point as a sequence by itself, whereas uniform convergence ties together the rate of convergence of sequences at each point x by requiring the same threshold N for all points x. Theorem 5.16 (Uniform Convergence Theorem). Suppose (f n ) is a sequence of functions f n : X Y, where X is a topological space and Y is a metric space. If (f n ) is a sequence of continuous functions that converges uniformly to a function f, then f is continuous. Example 5.3. Let X [0, 1] and Y R. Suppose f n : X Y is defined by f n (x) x n, where n N. Notice that (f n ) converges pointwise to f, where f(x) = 0 for all x [0, 1) and f(x) = 1, x = 1. Thus, a sequence of continuous functions converges to a discontinuous function. However, this sequence of functions does not converge uniformly. Proof. We shall demonstrate the pointwise convergence of (f n ) to f. Choose some x (0, 1) and define (a n ) by a n f n (x) = x n. Let ɛ > 0. Then, for all n > N log ɛ log x, xn 0 < ɛ and thus a n converges to 0. For x = 0 and x = 1 it is clear that (f n (x)) converges to f(0) and f(1), respectively. 6 Acknowledgements These notes greatly benefited from the notes of Kim Border at caltech.edu/~kcb/notes.shtml and from Appendix A of Daron Acemoglu s Introduction to Modern Economic Growth. I have also consulted James Munkres Topology. 24

22 7 References Simon, Blume: Ch. 12, 29 Acemoglu: Appendix A.1 5 Abbott, Stephen. Understanding Analysis. Springer-Verlag, New York Simon, Carl P., Lawrence Blume. Mathematics for Economists. Norton, New York Solow, Daniel. How to Read and Do Proofs. 3.ed. Wiley, New York