Math 395: Topology Bret Benesh (College of Saint Benedict/Saint John s University) October 30, 2012
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Contents Acknowledgments v 1 Topological Spaces 1 2 Closed sets and Hausdorff spaces 7 iii
iv CONTENTS
Acknowledgments I would like to thank the Educational Advancement Foundation and the Academy of Inquiry Based Learning for providing the grant that allowed me to create these notes. v
vi ACKNOWLEDGMENTS
Chapter 1 Topological Spaces Definition 1 A topology on a set X is a collection T of subsets of X such that: 1. X T and T 2. The finite intersection of sets in the topology is also a set in the topology. 3. The union of an arbitrary collection of sets in the topology is a set in the topology. In this case, X is called a topological space. Definition 2 Let X be a set with a topology T. A subset U X is said to be open if U T. Exercise 3 List all of the possible topologies of a two-element set {a, b}. Exercise 4 List all of the possible topologies of a three-element set {a, b, c}. Definition 5 The trivial topology of a set X is {, X}. Theorem 6 For any set X, the trivial topology is, indeed, a topology. Definition 7 The discrete topology of a set X is the collection of all subsets of X. Theorem 8 For any set X, the discrete topology is, indeed, a topology. 1
2 CHAPTER 1. TOPOLOGICAL SPACES Exercise 9 Determine the discrete topology of {a, b, c}. Definition 10 The cofinite topology of a set X consists of with the collection of all subsets of X that have finite complements. Exercise 11 Determine the cofinite topology of {a, b, c}. Theorem 12 For any set X, the cofinite topology is, indeed, a topology. Definition 13 Suppose that T and T are two topologies of a set X. If T T, then T is finer than T. If T is finer than T and T T, we say that T is strictly finer than T. Definition 14 Suppose that T and T are two topologies of a set X. If T T, then T is coarser than T. If T is coarser than T and T T, we say that T is strictly coarser than T. Exercise 15 Let X = {a, b, c} and T = {, {b}, {a, b}, {b, c}, X}. Find a topology of X that is finer than T. Exercise 16 Let X = {a, b, c} and T = {, {b}, {a, b}, {b, c}, X}. Find a topology of X that is coarser than T. Exercise 17 Let X = R and T = {, R} {(a, ) : a R}. Show that T is a topology of R. Definition 18 If X is a set, a basis for a topology on X is a collection B of subsets of X (each subset is called a basis element) such that 1. For all x X, there is at least one basis element B of B such that x B. 2. If x is contained in two basis elements B 1 and B 2, then there is a basis element B 3 such that B 3 B 1 B 2 and x B 3. Definition 19 Let B be a basis for a topology on a set X. The topology generated by B (call it T ) is determined by the following rule: A subset U of X is open if and only if for each x U, there is a basis element B B such that x B and B U.
Theorem 20 For any set X and any basis B for a topology on X, the topology generated by B is, indeed, a topology. Theorem 21 Let X = R and B = {(a, b) : a, b R}. Then B is a basis for a topology on R. Question 1 (Norby s First Question) Does there exist a topology of the real line such that the open sets of the topology are generated by the closed intervals of the form [a, b] for a, b R? Question 2 (Lane s First Question) Can a basis give rise to two different topologies? Theorem 22 Let X = R 2 and B be the set of circular regions (i.e. interiors of circles) in R 2. Then B is a basis for a topology on R 2. Theorem 23 Let X = R 2 and B be the set of rectangular regions (i.e. interiors of rectangles) in R 2. Then B is a basis for a topology on R 2. Theorem 24 Let X be any set and B = {{x} : x X}. Then B is a basis for the discrete topology on X. Theorem 25 Let X be a set and B a basis for a topology T on X. Then T equals the collection of all unions of elements of B. Theorem 26 Let X be a set, B a basis for a topology T on X, and B a basis for a topology T on X. Then the following are equivalent: 1. T is finer than T. 2. For each x X and each basis element B B containing x, there is a basis element B B such that x B B. Exercise 27 Prove that B from Theorem 22 generates the same topology as B from Theorem 23 Theorem 28 Let X be a topological space. Suppose that C is a collection of open sets of X such that for each x X and each open set U of X with x U, there is an element C C such that x C U. Then C is a basis for the topology of X. 3
4 CHAPTER 1. TOPOLOGICAL SPACES Notation 29 Let X = R and B be the set of all subsets of R of the form (a, b) for some a, b R. Theorem 21 states that B is a basis for a topology on R; this topology is called the standard topology; unless otherwise stated, this is the topology that is assumed when talking about R. Theorem 30 Let X = R and B be the set of all subsets of R of the form [a, b) for some a, b R. Then B is a basis for a topology on R (this topology is called the lower limit topology, and is denoted R l ). Theorem 31 The lower limit topology on R is strictly finer than the standard topology on R. Exercise 32 In the standard topology of R, show that there are elements of the topology that are not open intervals. Exercise 33 In the topology described in Theorem 22, show that there are elements of the topology that are not circles. Definition 34 For any set X containing at least two elements, and suppose that X is linearly ordered by some relation <). Define the following intervals: (a, b) = {x X : a < x < b} (this is an open interval) (a, b] = {x X : a < x b} (this is an half-open interval) [a, b) = {x X : a x < b} (this is an half-open interval [a, b] = {x X : a x b} )(this is an closed interval) Definition 35 Let X be a linearly ordered set with at least two elements, and let < be the ordering relation. Let B be the collection of all sets of the following types: 1. All open intervals (a, b) in X 2. All half-open intervals [a 0, b) in X, where a 0 is the smallest element in X. 3. All half-open intervals (a, b 0 ] in X, where b 0 is the largest element in X.
5 The topology generated by B is called the order topology. Problem 36 We have already seen an example of the order topology. Where did we see it? Theorem 37 The order topology of N is equal to the discrete topology. Definition 38 Let X be a linearly ordered set with at least two elements, and let a X. Define the following sets to be rays: 1. (a, ) = {x X : x > a} 2. (, a) = {x X : x < a} 3. [a, ) = {x X : x a} 4. (, a] = {x X : x a} The first two rays are called open rays; the last two are called closed rays. Theorem 39 Let X be a set with at least two elements and a linear ordering relation <. Then the open rays are, indeed, open sets in the order topology of X. Definition 40 Let X and Y be topological spaces. The product topology on X Y is the topology generated by. B = {U V : U is an open set in X and V is an open set in Y } Theorem 41 The set B from the previous definition is a basis for a topology. Problem 42 Show that while the B from the previous definition is a basis for a topology, it is not itself a topology. Theorem 43 Suppose B is a basis for a topology of X and C is a basis for a topology on Y. Then {B C : B B and C C} is a basis for X Y. Problem 44 Explain how the topology described in Theorem 23 is the product topology on R R, where R is considered to have the standard topology. (This topology on R 2 is called the standard topology on R 2 ).
6 CHAPTER 1. TOPOLOGICAL SPACES Definition 45 Let X be a topological space with topology T and Y a subset of X. Then T Y = {Y U : U T } is a topology on Y called the subset topology. In this case, we say that Y is a subspace of X. Theorem 46 For any set X and any subset Y, the subspace topology is, indeed, a topology. Theorem 47 If B is a basis for a topology on X, then B Y = {B Y : B B} is a basis for the subspace topology on Y. Definition 48 If Y is a subspace of X, we say that a set U is open relative to Y (or open in Y ) if it belongs to the topology of Y. We say that U is open relative to X (or open in X) if it belongs to the topology of X. Problem 49 Give an example of a topological space X, a subspace Y, and a set U Y such that U is open relative to Y, but U is not open relative to X. Theorem 50 Let Y be a subspace of X. If U is open relative to Y and Y is open relative to X, then U is open relative to X. Theorem 51 If X is a linearly ordered set with at least two elements in the order topology and Y is an interval or ray in X, then the subspace topology and the order topology in Y are the same. Theorem 52 If A is a subspace of X and B is a subspace of Y, then the product topology on A B is the same as the subspace topology on A B viewed as a subset of X Y. Question 3 (Humbert s First Question) Can we extend Theorem 43 to involve a direct product of n topological spaces? If so, prove it. If not, explain why we cannot.
Chapter 2 Closed sets and Hausdorff spaces Definition 53 A subset A of a topological space X is closed if X A is open. Theorem 54 In the standard topology of R, the interval [a, b] is closed. Theorem 55 In the standard topology of R, the subset {x} is closed for any x R. Problem 56 Find a subset of R (with the standard topology) that is neither closed nor open. Theorem 57 In R 2 with the standard topology, the set {(x, y) : x 0, y 0} is closed. Theorem 58 In the discrete topology of a set X, every subset is closed. Problem 59 Consider R with the standard topology, and let Y = [0, 1] (2, 3). Prove the following: 1. In the subspace topology of Y, the set [0, 1] is open. 2. In the subspace topology of Y, the set (2, 3) is closed. Theorem 60 Let X be a topological space. The following are true: 7
8 CHAPTER 2. CLOSED SETS AND HAUSDORFF SPACES 1. X and are closed. 2. Arbitrary intersections of closed sets are closed. 3. Finite unions of closed sets are closed. Definition 61 If Y is a subspace of X, we say that a set U is closed relative to Y (or closed in Y ) if A is closed in the subspace topology of Y. Theorem 62 Let Y be a subspace of X. Then a set A is closed in Y if and only if it equals the intersection of a closed set of X with Y. Theorem 63 Let Y be a subspace of a topological space X and A be a subset of Y. If A is closed in Y and Y is closed in X, then A is closed in X. Definition 64 The interior of A (denoted A ) is the union of all open sets contained in A. Definition 65 The closure of A (denoted A) is the intersection of all closed sets containing A. Problem 66 Consider R with the standard topology. Find the interior and closure of [0, 1). Problem 67 In the standard topology of R, find the closure of the subset {x} for any x R. Theorem 68 Let A be any subset of a topological space X. Then A A A. Theorem 69 Let A be any subset of a topological space X. If A is open, then A = A. Theorem 70 Let A be any subset of a topological space X. If A is closed, then A = A. Theorem 71 Let Y be a subspace of X, A be a subset of Y, and A the closure of A in X. Then the closure of A in Y is A Y. Definition 72 Let A and B be sets. We say A intersects B if A B is nonempty.
9 Theorem 73 Let A be a subset of X. Then: 1. An element x X is in A if and only if every open set U containing x intersects A. 2. If X has basis B, then x A if and only if every B B that contains x intersects A. Definition 74 Let X be a topological set, x X, and U X be open. Then if x U, we say that U is a neighborhood of x. Theorem 75 Let A be a subset of X. Then an element x X is in A if and only if every neighborhood of x intersects A. Exercise 76 Let X = {a, b, c} and T = {, X, {b}, {a, b}, {b, c}}. Find: 1. Every neighborhood of b. 2. Every neighborhood of a. Exercise 77 Consider R with the standard topology, and let A = (0, 1]. Describe A. Exercise 78 Consider R with the standard topology, and let B = { 1 n : n N}. Describe B. Exercise 79 Consider R with the standard topology, and let C = {0} (1, 2). Describe C. Exercise 80 Consider R with the standard topology. Describe N. Exercise 81 Consider R with the standard topology, and let Y = (0, 2] be a subspace with the subspace topology. Let A = (0, 1). Prove: 1. The closure of A in R is [0, 1]. 2. The closure of A in Y is (0, 1]. Definition 82 Let X be a topological set and A be a subset of X. Then we say a point x X is a limit point of A if every neighborhood of x intersects A in at least one point other than x.
10 CHAPTER 2. CLOSED SETS AND HAUSDORFF SPACES Theorem 83 Let X be a topological space and A a subset of X. Then x X is a limit point of A if it belongs to the closure of A {x}. Exercise 84 Let X be any topological space (with topology T ) and x be any element of X. Find all limit points of the subset {x}. Exercise 85 Let X = {a, b, c} and T = {, X, {b}, {a, b}, {b, c}}. Find: 1. The limit points of {a, b}. 2. The limit points of {a, c}. Exercise 86 Consider R with the standard topology, and let A = (0, 1]. Find the limit points of A. Exercise 87 Consider R with the standard topology, and let B = { 1 n : n N}. Find the limit points of B. Exercise 88 Consider R with the standard topology, and let C = {0} (1, 2). Find the limit points of C. Exercise 89 Consider R with the standard topology. Find the limit points of N. Exercise 90 Consider R with the standard topology, and let Y = (0, 2] be a subspace with the subspace topology. Let A = (0, 1). 1. Describe the limit points of A in R. 2. Describe the limit points of A in Y. Exercise 91 Give an example of topological space X, subset A, and limit point x of A such that x is not an element of A. Theorem 92 Let A be a subset of a topological space X. Let A be the set of all limits of A. Then A = A A. Theorem 93 A subset of a topological space is closed if and only if it contains all of its limit points.