Math 5801 General Topology and Knot Theory

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1 Lecture 3-8/27/2012 Math 5801 Ohio State University August 27, 2012 Course Info Textbook (required) J. R. Munkres, Topology (2nd Edition), Prentice Hall, Englewood Cliffs, NJ, ISBN-10: Reading for Wednesday, August 29 Chapter , pgs HW 2 for Wednesday, September 5 Chapter 1.4: 3a, 4a-b Chapter 1.5: 1 Chapter 1.6: 2, 7 Prove Propositions 10(1) and 11(3)

2 Zermelo-Fraenkel Axioms (EXT) (Extensionality Axiom) Sets are determined by their elements: For all sets A and B, (A = B) iff ( x, x A iff x B) (PAIR) (Pair Set Axiom) The set {x, y} exists: For all x and y, there is A s.t. z, [z A iff (z = x or z = y)] (UNION) (Axiom of Unions) Unions of sets exist: For all sets A there is a set B such that x B iff A A s.t. x A (POW) (Power Set Axiom) Every set has a power set: For all sets A there is a set P such that B, B P iff ( x B, x A) (SEP) (Separation Axiom) If A is a set and Q(x) is a logical statement which depends on x then the set of all elements of A satisfying Q(x) exists: A and statements Q(x), B s.t. x, x B iff x A and Q(x) is true. (REP) (Replacement Axiom) If A is a set and f is a function then B = f (A) is a set: For any set A and any statement Q(x, y) ( x A,!y s.t. Q(x,y) is true) implies ( B s.t. y B iff x A s.t. Q(x, y) is true) (EMP) (Existence of an empty set) There is a set which has no elements: There exists E s.t. x, x / E (REG) (Axiom of Regularity) Every nonempty set A contains an element x with x A =. For all sets A if A then x A s.t. ( y s.t. (y x) and (y A)) (INF) (Axiom of Infinity) There is an infinite set and it contains, { }, {, { }},. There exists S s.t. S and x S, x {x} S. (AC) (Axiom of Choice) Given a set A of nonempty disjoint sets there is a set C which contains exactly one element from each set in A. For all A s.t. ( / A) and ( A, B A,A B implies A B = ) C s.t. A A,!x C s.t. x C A. Forget everything you know about set theory. Can only talk about sets that exist due to ZFC axioms. Note: We can prove existence of some sets without Axiom of Choice (AC). Such sets are constructible. Using (AC) is fine but sets whose existence depends on (AC) are inaccessible.

3 Proposition 7 (Empty set is unique) Let denote the set given by (EMP). Then if the set A satisfies same properties as given in (EMP) then A =. Proof. By (EMP) there exists a set E such that x, x / E. Call it. Suppose there is A s.t. x, x / A. Then x, (x / A) and (x / ). Then x, (x A) and (x ). Then x, ( (x A) and (x ) ) or ( (x A) and (x ) ). Then x, (x A) iff (x ). So by (EXT) A =. Proposition 8 (Subsets of the empty set are empty) A iff A =. Proof. A = iff x, (x / A) iff x, (x A) iff x, (x A) implies (x ) iff A.

4 Proposition 9 P( ) = { }. Proof. By definition P( ) has property given in (POW). Namely B, B P( ) iff x B, x. By definition { } = {, } which by (PAIR) satisfies B, B {, } iff (B = ) or (B = ). Thus B, B { } iff B =. B P( ) iff x B, x iff B }{{} iff B = iff B { }. Thus B, B P( ) iff B { }. Thus by (EXT) we have P( ) = { }. Prop. 8 Proposition 10 (Miscellaneous properties of sets) If A, B and C are sets then: 1. A A. (Reflexivity) 2. A = B iff (A B and B A). 3. (A B and B C) implies A C. (Transitivity) 4. A A B. 5. A B A. 6. A. 7. A = A. 8. A =. 9. =. 10. B P(A) iff B A.

5 Proposition 11 (More properties of sets) 1. x {y} iff x = y. 2. {x} A iff x A. 3. {a, b} = {x, y} iff ((a = x and b = y) or (a = y and b = x)). Proposition 12 For any set A we have A / A. Proof. Let A be a set. Suppose A A. By (PAIR) the set {A} exists. A A and A {A} so A {A}. By (REG) there is x {A} such that x {A} =. But x {A} implies x = A implies x {A} = A {A} =. }{{} so A / A. Prop. 11(1) CONTRADICTION.

6 Ordered Pairs in ZFC Definition 13 (Ordered pair) (x, y) = {x, {x, y}}. Proposition 14 (x, y) = (z, w) iff x = z and y = w. Proof. Suppose x = z and y = w. Then {x, {x, y}} = {z, {z, w}}. Thus (x, y) = (z, w). Ordered Pairs in ZFC Proof of Prop. 14 (continued). Suppose (x, y) = (z, w). Then {x, {x, y}} = {z, {z, w}}. Then (x = z and {x, y} = {z, w}) or (x = {z, w} and {x, y} = z). CASE I: Suppose x = z and {x, y} = {z, w}. Then x = z and ((x = z and y = w) or (x = w and y = z)). If x = z and (x = z and y = w) then we are done. If x = z and (x = w and y = z) then w = x = z = y so (x = z and y = w) and we are done. CASE II: Suppose x = {z, w} and {x, y} = z. Then x = {{x, y}, w}. Let A = {{x, y}, x}. A fails (REG). CONTRADICTION

7 The Cartesian Product We d like to define A B = {(a, b) a A and b B}. In order to use (SEP) we need a set which contains each (a, b). Suppose a A and b B. By definition (a, b) = {a, {a, b}}. a A and {a, b} P(A B). So {a, {a, b}} A P(A B). So {a, {a, b}} P(A P(A B)). Definition 15 (Cartesian Product) A B = { C P(A P(A B)) } a A, b B s.t. C = (a, b). or more informally, A B = {(a, b) a A and b B}. Functions Definition 16 (Function) Let A and B be sets. A function f : A B is a pair f = (G, B) where G A B and for all a A there is a unique b B such that (a, b) G. If (a, b) G we write f (a) = b. The set dom(f ) = A is called the domain of f. The set codom(f ) = B is called the codomain or range of f. The set graph(f ) = G is called the graph of f. Observations: Let f be a function. Then a dom(f ) iff b B s.t. (a, b) graph(f ). If f and g are functions and f = g then graph(f ) = graph(g) and codom(f ) = codom(g). Moreover dom(f ) = dom(g). (Why?)

8 Functions Definition 17 Let f : A B be a function. 1. f is injective (or one-to-one) if 2. f is surjective (or onto) if x, y A, ( f (x) = f (y) implies x = y ). b B, a A s.t. f (a) = b. 3. f is bijective (or gives a one-to-one correspondence) if it is injective and surjective. 4. If f is bijective then its inverse is the function f 1 : B A with graph(f 1 ) = { (b, a) B A a A s.t. f (a) = b }. Functions Definition 18 (Composition of functions) Let f : A B and g : B C be functions. Then the composition of g and f is the function g f : A C with graph(g f ) = { (a, c) A C b B s.t. f (a) = b and g(b) = c }. Definition 19 (Identity function) The identity function on the set A is the function ida : A A with graph(ida) = { (a, a) A A a A }. Proposition 20 If f : A B is a bijection then f 1 f = ida and f f 1 = idb.

9 Functions Definition 21 Let f : A B be a function. 1. If X A then the image of X under f is the set f (X ) = { b B x X s.t. f (x) = b } 2. In particular f (A) is called the image of f. 3. If Y B then the preimage of Y under f is the set f 1 (Y ) = { a A y Y s.t. f (a) = y } 4. Let X A. The restriction of f to X is the function f X : X B where graph(f X ) = { (x, b) X B x X s.t. f (x) = b } Functions Proposition 22 Let f : A B be a function. 1. If X A then X f 1 (f (X )). 2. If Y B then f (f 1 (Y )) Y. 3. If f is injective and X A then X = f 1 (f (X )). 4. If f is surjective and Y B then f (f 1 (Y )) = Y.

10 Functions Definition 23 (The set of all functions) Let A and B be sets. The set of all functions from A to B is the set B A = { f f : A B }. More formally, B A = { f P(A B) {B} } f : A B. Let A be a nonempty set. Which of the following are empty? 1. A A 2. A 3. A 4. Relations Definition 24 (Relation) Let A be a set. A relation is a subset C A A. We write xcy instead of (x, y) C. Example 25 If f : A A then graph(f ) is a relation on A.

11 Equivalence Relations Definition 26 (Equivalence Relation) Let A be a set. An equivalence relation on A is a relation C on A satisfying: 1. (Reflexivity) xcx for all x A. 2. (Symmetry) If xcy then ycx. 3. (Transitivity) If xcy and ycz then xcz. We ll usually write x y instead of xcy. Definition 27 (Equivalence Class) Let be a relation on A. If x A then the equivalence class of x is the set Ex = {y A x y} Partitions Definition 28 (Partition) Let A be a set. A partition of A is a subset D P(A) with 1. D = A. 2. If X, Y D then (X Y = X = Y ). Proposition 29 Let A be a set with an equivalence relation and let E = { E P(A) x A s.t. E is equiv. class of x }. Then E is a partition of A.

12 Order Relations Definition 30 (Order Relation) Let A be a set. An order relation (or simple order or linear order) on A is a relation C on A satisfying: 1. (Comparability) For all x, y A, x y implies (xcy or ycx). 2. (Nonrefexivity) For all x A, xcx. 3. (Transitivity) If xcy and ycz then xcz. We ll usually write x < y or y > x instead of xcy. Definition 31 (Open Interval) Let A be a set with an order relation < and let a, b A. The set Is an open interval in A. {x A a < x < b} Order Relations Definition 32 (Bounds) Let A be a set with an order relation < and let X A be a subset. 1. An upper bound for X is an elt. a A s.t. for all x X, x a. 2. X is bounded above if it has an upper bound. 3. X has a largest element if x X s.t. x is an upper bound of X. 4. A lower bound for X is an elt. a A s.t. for all x X, a x. 5. X is bounded below if it has a lower bound. 6. X has a smallest element if x X s.t. x is a lower bound of X. 7. The supremum of X sup X is the smallest upper bound of X. 8. The infimum of X inf X is the smallest upper bound of X.