f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write


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1 Math 525 Chapter 1 Stuff If A and B are sets, then A B = {(x,y) x A, y B} denotes the product set. If S A B, then S is called a relation from A to B or a relation between A and B. If B = A, S A A is called a relation on A. For a relation S A B, we define Dom S = {x A y B s.t. (x,y) S} Ran S = {y B x A s.t. (x,y) S} domain of S range or image of S S 1 = {(y,x) (x,y) S} inverse of S Note that S 1 B A is a relation from B to A. Clearly, DomS 1 = Ran S and Ran S 1 = Dom S. If S A B and T B C, we define the composition T S A C by T S = {(x,y) z B s.t. (x,z) S and (z,y) T }. It should be clear that T S = φ if (Ran S) (Dom T) = φ. If S A B, C A and D B, we define: S(C) = {y B x C s.t. (x,y) S} image of C relative to S S 1 (D) = {x A y D s.t. (x,y) S} preimage of D relative to S. In the case that C or D is a singleton set, the { } are often omitted in the above definitions: S(x) means S({x}) and S 1 (y) means S 1 ({y}). Functions If A, B are sets, f : A B is a function from A into B if f A B, Dom f = A, and f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write f(x) = y instead. In general, B is called the codomain of f and Ran f B. If f is a function satisfying f(x) = f(y) = x = y for all x,y A, then f is injective.
2 If Ran f = B, then f is surjective (relative to the codomain B). A function f : A B which is both injective and surjective is said to be bijective (or a bijection). Since a function f : A B is a special case of a relation from A to B, inverses and compositions are defined for functions as described previously. It is easy to check that if f : A B and g : B C are functions, then g f : A C is also a function. Furthermore, f 1 B A always exists as a relation, but f 1 is not necessarily a function. Note that f 1 : Ranf A is a function (indeed, a bijection) iff f is injective. Also, f 1 : B A is a bijection iff f : A B is a bijection. Proposition A1 Let f : A B and g : B A be functions such that g f = i A and f g = i B. Then f and g are bijections, and f = g 1. Equivalence Relations In studying equivalence and order relations, we restrict our attention to relations of the form S A A. In other words, S is a relation on A. We define a relation S on A to be: (1) Reflexive iff x A, (x,x) S; (2) Symmetric iff x,y A, (x,y) S = (y,x) S; (3) Transitive iff x,y,z A, ((x,y) S and (y,z) S) = (x,z) S. Another notion closely related to an equivalence relation on A is a partition of a set A. Let C = {A i i I}, where I is an arbitrary index set and A i A for all i I. Then C is called a partition of A iff C = i I A i = A and A i A j = φ whenever i j in I. Equivalence relations and partitions are essentially the same thing. If S is an equivalence relation on A, then for each x in A, [x] = {y x y} is called the equivalence class containing x, and the set A/S = {[x] x A} of equivalence classes forms a partition of A, which is called the quotient set of A relative to S. On the other hand, if C = {A i i I} is any partition of A, then the relation S = {(x,y) S i I s.t. x,y A i } is an equivalence relation on A and A/S = C.
3 A relation S on a set A is: antisymmetric iff x,y A, (x,y) S and (y,x) S x = y; comparable iff x,y A, either (x,y) S or (y,x) S. A relation S on A is called a partial order relation (or just a partial order) iff it is reflexive, transitive, and antisymmetric. A partial order which is comparable is called a total order (other terms used for this same concept include simple order, linear order, and, in Munkres, order). A partially ordered set is often called a poset; a simply ordered set is often called a chain. In dealing with any type of order relation S on A, it is common to write x y or y x rather than (x,y) S. If x y and x y, then we write x < y (or y > x), which is referred to as a strict inequality. Using the inequality notion, we can define a poset (A, ) to be a set A with a partial order satisfying (1) x x for all x A (reflexive) (2) (x y and y z) = x z for all x,y,z A (transitive) (3) (x y and y x) = x = y for all x,y A (antisymmetric) (A, ) is a totally ordered set or chain iff, in addition, (4) x,y A, either x y or y x. Examples 1. R, Q, Z and N are all totally ordered sets, with their usual orderings. 2. Let X be any set and let P(X) denote the power set of X (the set of all subsets of X). Define an ordering on P(X) as follows: For all B,C P(X), B C iff B C. Then (P(X), ) is a poset, but is not simply ordered if X has 2 or more elements. For if x,y X and x y, then the sets {x} and {y} do not satisfy the comparability condition, since {x} {y} and {y} {x}.
4 3. Consider two frequently used partial orders on the Euclidean plane R 2 : (1) (a,b) (c,d) iff a c and b d (pointwise order) (2) (a,b) (c,d) iff a < c or a = c and b d (dictionary order) Relative to pointwise ordering, R 2 is a poset but is not simply ordered, since (for example) the elements (1, 2) and (2, 1) are not comparable. Relative to the dictionary order, R 2 is totally ordered. Let (A, ) be a poset and φ B A. Then y A is an upper bound for B iff b y for all b B. Let U(B) be the set of all upper bounds of B. If U(B) has a least element (i.e. x U(B) such that x y for all y U(B)), then x is the least upper bound of B, denoted supb. Analogously, we can define the greatest lower bound of B, which, if it exists, is denoted by inf B. Note that supb will fail to exist either if U(B) is empty or if U(B) φ but contains no least element. A poset in which each set bounded above has a least upper bound and each set bounded below has a greatest lower bound is defined to be order complete. Note that R, N, Z, P(X) and (R 2, pointwise order) are order complete, whereas Q and (R 2, dictionary order) are not. An element x in a poset (A, ) is a maximal element iff y A such that x < y. If A contains a greatest element, that element is the unique maximal element. But a maximal element need not be the greatest element. For example, let A = {(x,y) R 2 y x} and use the pointwise order on A. Then every element on the line y = x is a maximal element of A, but A has no greatest element. Axiom of Choice If C = {A i i I} is an arbitrary nonempty collection of nonempty sets, there is a function f : I C such that f(i) A i for all i I. Many constructions and proofs in topology involve selecting an arbitrary element from each member of an arbitrary collection of nonempty sets, and the Axiom of Choice justifies this procedure. The Axiom of Choice does not explicitly describe how to carry out this selection;
5 rather it asserts that the assumption that such a selection can be made will not lead to any contradiction. A small group of mathematicians called Constructionists reject this axiom, but most mathematicians accept the Axiom of Choice because of its many useful and interesting applications. Next, we state Zorn s Lemma, which is (not obviously) equivalent to the Axiom of Choice. Zorn s Lemma Let Z be a nonempty poset in which each simply ordered subset has an upper bound. Then Z contains a maximal element. Zorn s Lemma can often be used to prove results to which the original statement of the Axiom of Choice does not seem applicable. To illustrate the use of Zorn s Lemma and some of the order properties discussed earlier, we ll next prove a theorem from linear algebra. Recall that a set B is a basis for a vector space V iff B is an independent set in V which spans V. Theorem A2 Every vector space V {0} has a basis. Proof: Let Z be the set of all linearly independent subsets of V, partially ordered by set inclusion. Since V {0}, there is a nonzero element x in V, and so {x} Z, and Z φ. To show that Z satisfies the hypothesis of Zorn s Lemma, let {D i i I} be a simply ordered (i.e. nested) subset of Z. Let D = D i. Because the D i are nested, it can be verified that i I D is linearly independent and hence an upper bound of {D i i I} in Z. Now, by Zorn s Lemma, Z must have a maximal element. That is, Z must contain a maximal independent subset of V ; call it B. To show that B is a basis for V, it remains only to show that B spans V. Let Y be the vector subspace of V spanned by B (sometimes written as Y = span B). If Y V, then y V Y. Since y is not in the span of B, the set B {y} would be an independent set in V which properly contains B, contradicting the maximality of B in Z. Thus Y = V, and B is a basis for V.
6 To form set exponents, it is useful to first extend the definition of set products. Let {X i i I} be an arbitrary collection of sets, and let P = {f : I X i f(i) Xi for all i I}. i I P is called the Cartesian product of the X i, denoted by P = X i. Note that each i I function f P is a function which chooses a representative element, f(i), from each set X i. Such a function is often referred to as a choice function, and its existence is a consequence of the Axiom of Choice. Also note that P = φ iff X i = φ for some i I. In dealing with set products, it is common to write f i I X i as f = (x i ) i I, where f(i) = x i. If I = {1, 2,...,n}, then f would be the ntuple (x 1,x 2,...,x n ). If I = N, then f would be the sequence (x n ) n N. Set exponentiation is simply a special case of set product. If A and I are sets, and X i = A for all i, then i I X i can be thought of as the product of A with itself I times, and we write i I X i = i I A = AI. By the earlier definition of i I X i, it follows that A I is simply the set of all functions from I into A. If A and I are finite sets with n and k elements, respectively, then A I contains n k elements. Note that R n = R I, where I = {1, 2,...,n}, and R N is the set of all (infinite) sequences of real numbers. The power set P(X) of all subsets of X is often written via exponential notation. Let 2 denote the set {0, 1}, and for any subset A of X, let χ A denote the characteristic function of A: 1, x A χ A (x) = 0, x X A. Then every f 2 X is the characteristic function of some subset of X, and each subset of X is uniquely defined by its characteristic function. Thus P(X) can be identified with 2 X.
7 Cardinal Numbers The Cardinal numbers are the equivalence classes induced by the relation of bijectivity on the class of all sets. The the cardinal number (equivalence class) of φ is called 0 ; we write φ = 0. The cardinal number (equivalence class) of every singleton set is called 1 ; we write {a} = 1. The cardinal number (equivalence class) of every nelement set is called n ; for example, {a,b,c,d,e} = 5. The cardinal number of N, Z, Z + and Q is called ℵ 0. ℵ 0 is the smallest infinite cardinal. A set is countable iff its cardinal number is ℵ 0, and is otherwise uncountable. The cardinal numbers are totally ordered by. Well Ordered Sets The theory of well ordered sets provides the foundation for the study of ordinal numbers. A well ordered set (w.o.set) is a poset in which every nonempty subset contains a least element. Proposition A3 Every wellordered set is totally ordered. Proof: Let X be a well ordered set, and let x,y X. Then the subset {x,y} must contain a least element, which is either x or y. If it is x, then x y. If it is y, then y x. Thus X is a poset which also satisfies the comparability axiom, and is therefore totally ordered. Every w.o.set contains a least element, but not every totally ordered set with a least element is well ordered. For example, [0, 1] (with the usual ordering) has a least element 0, but the subset (0, 1) does not. Every finite totally ordered set is well ordered, as is the set N of natural numbers. If A and B are disjoint well ordered sets, then A B can be made into a w.o.set by stipulating that (a A and b B) = a < b, while leaving the orders within A and B unchanged. For instance, suppose A = {a 0,a 1,...,a n,...} and B = {b 0,b 1,...,b n,...}, where the elements of these sets are listed in increasing order from left to right. Then A B = {a 0,a 1,...,a n,...,b 0,b 1,...,b n,...}, with elements again listed in increasing order from left to right, is a w.o.set. Furthermore, for any w.o.sets A and B, the set A B is well ordered relative to the dictionary order. Any subset of a well ordered set with the inherited
8 order is also well ordered. In a totally ordered set X, b is the immediate successor of a iff a < b and a < c b = c = b. The immediate predecessor is defined similarly. Proposition A4 In a w.o.set X, every element which is not the greatest element of X has an immediate successor, but every element does not necessarily have an immediate predecessor. Proof: Let a be an element of X which is not the greatest element. And let b be the least element of {x A a < x}, which is is nonempty. Then b is the immediate successor of a. In the example mentioned previously involving A B, neither a 0 nor b 0 has an immediate predecessor. Proposition A5 In a well ordered set X, every nonempty subset bounded above has a least upper bound, and every nonempty subset has a greatest lower bound. Proof: If A φ is bounded above, then the set U(A) of all upper bounds of A contains a least element, which is the l.u.b of A. Also, every A φ contains a least element, which is the g.l.b. of A. An important theorem in set theory is the Well Ordering Theorem, first proved by Zermelo in 1904 using the Axiom of Choice. It was later discovered that the Well Ordering Theorem is equivalent to the Axiom of Choice. We ll state the Well Ordering Theorem here without proof. Well Ordering Theorem Every set can be well ordered. This theorem is clear for countable sets, but is not obvious in general. Like other results based on the Axiom of Choice, the Well Ordering Theorem asserts the existence of something without providing a means for achieving it. If (X, ) is a w.o.set and x X, we define S x = {y X y < x}. S x is called the initial section of X determined by x. Recall from a previous example the wellordered set A B = {a 0,a 1,...,a n,...,b 0,b 1,...,b m,...,...}, formed from the union of two disjoint wellordered sets A and B. For this set, we have
9 S a0 = φ, S a2 = {a 0,a 1 }, S an = {a 0,a 1,...,a n 1 }, S b0 = A, etc. Proposition A6 There exists an uncountable wellordered set S Ω such that every initial section of S Ω is countable. Proof : Start with any uncountable set X and use the Well Ordering Theorem to wellorder X. Let Y be another uncountable wellordered set obtained by setting Y = X X with the dictionary order. If x 1 is not the least element of X, then the initial segment of Y determined by (x 1,x 1 ) is clearly uncountable, and so A = {y Y S y is uncountable } is a nonempty set in Y, which by assumption contains a least element; call it Ω. Note that by assumption, S Ω S y for all y Y. Claim 1: Every initial segment of S Ω countable. If not, then there would exists a z < Ω such that the initial section (S Ω ) z = Y z is uncountable. By definition of A and Ω, we d then have z A and z Ω, which is a contradiction. Claim 2: S Ω is uncountable. This follows immediately, since Ω A. This uncountable w.o.set S Ω plays an important role as a source of examples in topology. Ordinal Numbers Two wellordered sets A and B are similar (we ll write A B) iff there is an orderpreserving bijection ψ from A to B. This is a bijection ψ : A B such that x y in A iff ψ(x) ψ(y) in B. Note that similar w.o.sets have the same cardinal number. If there is a b B such that A B b, we write A B. Ordinal numbers are defined to be the equivalence classes obtained when the class W of all wellordered sets is partitioned into equivalence classes relative to the equivalence relation similarity. If L W, let L denote the ordinal number of the equivalence class containing L. As before, we denote by L the cardinal number for L. For each ordinal number λ, there is an associated cardinal number λ, which is the cardinality of any set L λ.
10 Let O denote the class of ordinal numbers. A total order on O is defined by λ µ iff there exists L λ and M µ such that L M or L M, with λ < µ corresponding to L M. The finite ordinal numbers are those ordinals λ for which λ is finite. All wellordered sets with exactly n elements are similar to each other, and thus they all belong to the same equivalence class in W. So there is a unique ordinal number corresponding to an nelement set, which it is natural to denote by n. Since there is an obvious orderpreserving bijection between finite ordinals and finite cardinals, it is customary to make no distinction between finite ordinal numbers and finite cardinal numbers. When we consider the infinite ordinals (those ordinals λ for which λ ℵ 0 ), the situation is drastically different. We define denumerable ordinals (resp., countable ordinals) to be those of cardinality ℵ 0 (resp., ℵ 0 ). We can consider the denumerable ordinals to correspond to the distinct ways to wellorder a denumerable set. For instance, N is already wellordered in its natural ordering. Some other w.o.sets defining denumerable ordinals are listed here: ω = {a 0,a 1,...,a n,...} = N, ω + 1 = {a 0,a 1,...,a n,...,b 0 } ω + ω = {a 0,a 1,...,a n,...,b 0,b 1,...,b m,...} ω ω = N N, with dictionary order The number of denumerable ordinals in uncountable. It is customary to refer to S Ω as the set of countable ordinals, and Ω = S Ω is called the first (least) uncountable ordinal. ω = N is the smallest infinite ordinal. Two important facts about ordinal numbers which we may use but will not prove are the following. Theorem A7 Any set of ordinals in their natural order is a wellordered set. Corollary Any set of cardinal numbers in their natural order is wellordered. The assertion that R = 2 ℵ 0 is the immediate cardinal successor of ℵ 0 is called the Continuum Hypothesis. This hypothesis, like the Axiom of Choice, can be accepted or rejected as an axiom of set theory.
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