notes1_intro

Topology MATH 4181 Fall Semester 1999 David C. Royster UNC Charlotte September 2, 1999 Chapter 1 Introduction to Topology 1.1 History1 Topology is thought of as a discipline that has emerged in the twentieth century. There are precursors of topology dating back into the 1600's. Gottfried Wilhelm Leibniz (1646{1716) was the rst to foresee a geometry in which position, rather than magnitude was the most important factor. In 1676 Leibniz use the term geometria situs(geometry of position) in predicting the development of a type of vector calculus somewhat similar to topology as we see it today. The rst practical application of topology was made in the year 1736 by the Swiss mathematician Leonhard Euler (1707{1783) in the K onigsberg Bridge Problem. Carl F. Gauss (1777{1855) predicted in 1833 that geometry of location would become a mathematical discipline of great importance. His studey of closed surfaces such as the sphere and the torus and surfaces much like those encountered in multi- dimensional calculus may be considered as a harbinger of general topology. Gauss was also interested in knots, which are of current interest today in topology. The word topology was rst used by the German mathematics Joseph B. Listing (1808{1882) in the title of his book Vorstudien zur Topologie (Introductory Studies in Topology ), a textbook published in 1847. Listing book dealt with knots and sur- faces but failed to generate much interest in either the name or the subject matter. Throughout much of the nineteenth and early twentieth centuries, much of what now falls under the auspices of topology was studied under the name of analysis situs (analysis of position). Bernard Riemann (1826{1866) was the rst mathematician to foresee topology in the generality it has achieved today. He initiated the study of connectivity of a surface, or the arrangement of holes in a surface. He used concepts in which the number of dimensions exceeded three, which at that time was generally conceded to be the maximum number of dimensions involved with any geometric object. Present-day topology can be traced to two primary sources: the development of non-euclidean geometry and the process of putting calculus on a rm mathematical foundation. 1The information here is taken from Principles of Topology by Fred H. Croom. 2 1.2. SETS AND SET OPERATIONS 3 1.2 Sets and Set Operations We need some basic information about sets in order to study the logic and the ax- iomatic method. This is not a formal study of sets, but consists only of basic deni- tions and notation. Bracesfandgare used to name or enumerate sets. The roster method for naming sets is simply to list all of the elements of a set between a pair of braces. For example the set of integers 1, 2, 3, and 4 could be named f1;2;3;4g: This does not work well for sets containing a large number of elements, though it can be used. The more common method for this is known as the set builder notation . A property is specied which is held by all objects in a set. P(x), read P of x , will denote a sentence referring to the variable x. For example, x= 23 xis an odd integer. 1x4. The set of all objects xsuch thatxsatisesP(x) is denoted by fxjP(x)g: The setf1;2;3;4gcan be named fxj1x4; x2Zg=fx2Zj1x4g: From hence forth, the words object, element , and member mean the same thing when referring to sets. Sets will be denoted mainly by capital Roman letters and elements of the sets by small letters. The following have the same meaning: a2A ais in setA ais a member of set A ais an element of set A Likewise,a62Ameans that aisnotan element of set A. Ais a subset ofBif every element of Ais also an element of B. The following have the same meaning: AB Every element of Ais an element of B c 1999, David Royster Introduction to Topology For Classroom Use Only 4 CHAPTER 1. INTRODUCTION TO TOPOLOGY Ifa2A, thena2B Ais included in B BcontainsA Ais a subset of B Note that a set is always a subset of itself. IfAandBare sets, then we say that A=BifAandBrepresent the same set: A=B AandBare the same set AandBhave the same members ABandBA The set which contains no elements is known as the empty set , and is denoted by ;. Note that for each set A,;A. The intersection of two sets AandBis the set of all elements common to both sets. The intersection is symbolized by A\Borfxjx2Aandx2Bg. The union of two sets AandBis the set of elements which are in AorBor both. The union is symbolized by A[Borfxjx2Aorx2Bg. 1.2.1 Universal Sets and Compliments When we are working in an area or on a certain problem, we always have a frame of reference in which we are working called a universal set . In our geometry course, it will be the set of points that lie on a plane. In calculus we consider the set of real numbers, the set of real functions, the set of dierentiable functions, and the set of continuous functions as universal sets. The complement of a setAis dened to be the set of all elements of the universal set which are not in A, and is symbolized by CA=A0=Ac. Note that A[Acis always the universal set, while A\Ac=;. The set dierence of the setsAandBis dened to be all of those elements in A which are not in B. It is denoted by AnB=fx2Ajx3Bg: Note thatAnBandBnAwill usually be dierent, and that even though AnB=; it need not follow that A=B. c 1999, David Royster Introduction to Topology For Classroom Use Only 1.3. PRODUCTS SETS 5 1.3 Products Sets LetXandYbe sets. The set of all ordered pairs f(x;y)jx2X and y2Ygis the product set XY, orCartesian product ordirect product ofXandY. Aslice of this product set isfxgYorXfygfor a given x2Xory2Y. Examples of common product spaces are the plane R2=RR, 3-space R3=RR2, a right circular cylinder,S1[0;1], or the torus, S1S1. Theorem 1.1 LetXandYbe sets and let A;CXandB;DY. a)A(B\D) = (A\B)(A\D). b)A(B[D) = (A[B)(A[D). c)A(YnD) = (AY)n(AD). d)(AB)\(CD) = (A\C)(B\D). e)(AB)[(CD)(A[C)(B[D). f)(XY)n(AB) = (X(YnB))[((XnA)Y) The concept of the product of two sets can be extended to more than two factors. IffXign i=1is a nite collection of sets, then their product is X1X2


Xn=nY i=1Xi=f(x1;x2;:::;xn)jxi2Xifor eachi= 1;2;:::;ng: For an innite collection of sets, the product is dened by 1Y i=1Xi=f(x1;x2;x3;:::)jxi2Xifor eachi= 1;2;:::g: 1.4 Functions Afunctionf:X!Yis a rule which assigns to each x2Xa uniquey2Yand we sayy=f(x). Ify=f(x) thenyis called the image ofxandxis called the preimage ofy. The setXis the domain offandYis the range orcodomain off. LetAX. The setf(A) =fy2Yjy=f(x) for somex2Agis called the image ofA. The setf(X) is called the image off. ForBY, the set f1(B) =fx2Xjf(x)2Bg is the inverse image ofBunderf. The set of points =f(x;f(x))2XYjx2Xg is called the graph of the function f. c 1999, David Royster Introduction to Topology For Classroom Use Only 6 CHAPTER 1. INTRODUCTION TO TOPOLOGY A functionf:X!Yisinjective if for distinct elements x1;x22X,f(x1)6=f(x2) inY. Another way to think of this is to say that fis injective if f(x1) =f(x2) implies thatx1=x2. Iff(X) =Y, the function fis said to be surjective . A function that is surjective and injective is called a bijection . In this case we have thatf:X!Yis a bijection provided that each member of Yis the image underfof exactly one member of X. In this case the inverse function f1:Y!X exists assigning to each element y2Yits unique preimage x=f1(y) inX. The identity function iX:X!Xfrom a set Xto itself is the function dened byiX(x) =xfor allx2X. This function is often denoted by 1 X. Iff:X!Yandg:Y!Zare functions, then the composite function gf:X! Zis dened by gf(x) =g(f(x)), forx2X. Denition 1.1 LetXbe a set. A sequence inXis a function f:Z+!Xwhose domain is the set of all positive integers, Z+or the set of positive integers less than or equal to some given positive integer N. The sequence is called nite if its domain isf1;2;:::;Ngand innite if its domain is all positive integers. 1.5 Equivalence Relations LetXbe a set. A relationRonXis a subset of XX. If (x;y)2Rwe will say thatxis related to ybyRand to write xRy. A relationRon a setXis called re exive ,symmetric , ortransitive if it satises the corresponding property below. (a)The Re exive Property :xRx for allx2X. (b)The Symmetric Property : IfxRy, thenyRx. (c)The Transitive Property : IfxRy andyRz, thenxRz. Denition 1.2 Anequivalence relation on a setXis a relation on Xwhich is re exive, symmetric, and transitive. Denition 1.3 Letdenote an equivalence relation on X. Forx2Xthe set [x] =fy2Xjyxgis calle dth e equivalence class of x. Proposition 1.1 LetXbe a set and letdenote an equivalence relation on X. a)x2[x]for eachx2X. b)xyif and only if [x] = [y]. c)x6yif and only if [x]\[y] =;. d) Forx;y2X,[x]and[y]are either identical or disjoint. c 1999, David Royster Introduction to Topology For Classroom Use Only 1.6. CARDINALITY 7 1.6 Cardinality We are often interested in how big sets are in relation to one another. Clearly, we can tell the dierence in sizes of two nite sets, but how do we dierentiate between two innite sets? Are there dierent sizes of innite sets? How do we compare sets to tell if one has a greater number of members? Denition 1.4 a setAisnite ifAis empty or if there is a bijection between A and the set of integers from 1toNfor some positive integer N. In the latter case, A is said to have Nmembers. If a set is not nite, it is called innite . Denition 1.5 A set is denumerable orcountably innite if there is a bijection between the set and the positive integers. A set which is either nite or denumerable is called countable . A set which is not countable is called uncountable . Lemma 1.1 a) Each subset of a nite set is nite. b) Each subset of a countable set is countable. c) Each set which contains an innite set is innite. d) Each set which contains an uncountable set is uncountable. Example 1.6.1 1. The setZ+[f0gof all non-negative integers is countable. The bijection is f:Z+[f0g!Z+given by f(n) =n+ 1;n2Z+[f0g: 2. The set of all integers, Zis countable. The bijection g:Z!Z+is given by g(n) =( 2n1 ifnis positive 2n ifnis negative 3. The product set Z+Z+is countable. One method is to use the Cantor Diagonalization Method to count the ordered pairs ( m;n). A second method is to dene the function g:Z+Z+!Z+by g(m;n) = 2m3n;(m;n)2Z+Z+: Now,gis not surjective, but the Fundamental Theorem of Arithmetic on the unique factorization into primes guarantees that the function gis injective. Thus, there is a bijection from Z+Z+to a subset of Z+. Since every subset of a countable set is countable, we have that Z+Z+is countable. c 1999, David Royster Introduction to Topology For Classroom Use Only 8 CHAPTER 1. INTRODUCTION TO TOPOLOGY Theorem 1.2 a) IffAigN i=1is a nite collection of nite sets, then bothSN i=1AiandQN i=1Aiare nite. (Finite unions and nite products of nite sets are nite.) b) IffAig1 i=1is a countable collection of countable sets, thenS1 i=1Aiis countable. (Countable unions of countable sets are countable.) c) IffAigN i=1is a nite collection of countable sets, then bothQN i=1Aiis countable. (Finite products of countable sets are countable.) Why didn't we claim that a countable product of countable sets is countable? Mainly because it is not true, as is seen in the following example. Example 1.6.2 LetAi=f0;1gfori= 1;2;:::. Let U=1Y i=1Ai=f(a1;a2;a3;:::)jai= 0 or 1g: Assume that Uis countable. Then there is a bijection f:Z+!U. For an element a= (a1;a2;a3;:::)2Uwe shall refer to a1as the rst coordinate, a2as the second coordinate, and so forth. We can list all of the elements in Uusing the bijection f. They areff(1);f(2);F(3);:::g. Consider the following element in U. Denex= (x1;x2;x3;:::) as follows: xi=( 0 if theithcoordinate of f(i) is 1 1 if theithcoordinate of f(i) is 0 Then, we have that for each positive integer i,x6=f(i) for they dier in the ith coordinate. This means that xis not in the exhaustive list of elements we have in our bijection. That is this bijection is not surjective. This contradiction show us that U cannot be countable. Theorem 1.3 The set of rational numbers is countable. Proof: There are several ways of proving this. One method is to use the Cantor Diagonalization Method to count the rationals. (Method 1 ) List the positive rational numbers in rows where the rst row consists of all positive rational numbers with 1 as a denominator, the second row lists all positive rational numbers with 2 as a denominator, and so on: 1 12 13 14 1::: 1 22 23 24 2::: 1 32 33 34 3::: 1 42 43 44 4::: ............... c 1999, David Royster Introduction to Topology For Classroom Use Only 1.6. CARDINALITY 9 Clearly, we have each positive rational number in here numerous times, but the diagonalization method will still show that there are a countable number of elements in this array. The positive rationals form a subset of this array, thus there must be a countable number of positive rationals. This will yield that there are a countable number of rational numbers. (Method 2 ) Every rational number can be expressed uniquely in lowest terms as m=n wheremandnare integers with no common positive divisor other than 1, and nis positive. Consider the function m=n7!(m;n) from the set of rational numbers intoZZ. This function is injective since the ordered pair ( m;n) determines only one rational number m=n. Thus, the set of rational numbers is equivalent to a subset of the countable set ZZ, and is hence countable. Theorem 1.4 The set of real numbers is uncountable. Proof: We will make use of the example of the countable product above. Each element inQ1 i=10;1 is a sequence consisting of 0's and 1's. Each of these sequences represents a unique real number between 0 and 1, by the correspondence (a1;a2;a3;:::)7!0:a1a2a3:::: This is a one-to=one correspondence. Thus, the set of real numbers between 0 and 1 representable as a decimal using only 0's and 1's is an uncountable set. Thus, R contains an uncountable set and hence is uncountable. Theorem 1.5 The set of irrational numbers is uncountable. Proof: Since the set of real numbers is the union of the set of rational numbers and the set of irrational numbers, if the set of irrationals were countable, then we would have that the real numbers are countable. That failing to be true, implies that the irrationals must be uncountable. c 1999, David Royster Introduction to Topology For Classroom Use Only Chapter 2 Metric Spaces 2.1 Denition and Some Examples Denition 2.1 LetXbe a set and d:XX!R+a function satisfying the following properties. For all x;y;z2X, a)d(x;y) = 0 if and only if x=y. b)d(x;y) =d(y;x). c)d(x;z)d(x;y) +d(y;z). Thendis called a metric ordistance function onXandd(x;y)is called the distance fromxtoy. The setXwith a metric dis called a metric space and is denoted by (X;d). Note that these properties are modeled on the distance functions that we have on RandR2. Doing so we usually call property (c) the Triangle Inequality . Example 2.1.1 The real line, Ris a metric space using the standard distance func- tion, the absolute value: d(a;b) =jabj. The above properties are standard proofs about the absolute value function. Example 2.1.2 The plane,R2, with the usual Euclidean distance formula is a metric space. IfP= (x1;y1) andQ= (x2;y2), then d(P;Q) =p (x2x1)2+ (y2y1)2: Example 2.1.3 These are special cases of the general Euclidean n-space, Rn=f(a1;a2;::: ;an)jai2Rg: 10 2.1. DEFINITION AND SOME EXAMPLES 11 The distance formula here is the usual distance formula for Euclidean n-space: d((x1;x2;:::;xn);(y1;y2;::: ;yn)) = nX i=1(xiyi)2!1=2 : dis called the usual metric onRn. To show that dis a metric, we need two standard results about vectors in Rn. First, leta2Rn. The normkakis the distance from ato the origin O= (0;0;:::; 0): kak=d(a;O) = nX i=1a2 i!1=2 : Theorem 2.1 (Cauchy-Schwarz Inequality) For any points a;b2Rn ja
bjkakkbk: Theorem 2.2 (The Minkowski Inequality) For any points a;b2Rn ka+bkkak+kbk: The distance between two points is given by d(a;b) =kabk. The rst two conditions making da metric are easily seen to be satised. We only need check the Triangle Inequality. Let x;y;z2Rn d(x;z) =kxzk=kxy+yzk kxyk+kyzk =d(x;y) +d(y;z) Example 2.1.4 [The Taxicab Metric] Dene a function d0:R2R2!Ras follows. Ifx= (x1;x2) andy= (y1;y2), then d0(x;y) =jx1y1j+jx2y2j: This is called the taxicab metric because the distance is measured along line segments parallel to the coordinate axes. Clearly,d0(x;x) = 0 and if d0(x;y) = 0, thenjx1y1j+jx2y2j= 0 which meansjx1y1j= 0 andjx2y2j= 0. This implies that x1=y1andx2=y2, andx=y. Because of the basic properties of the absolute value, it is obvious that d0(x;y) =d0(y;x). The Triangle Inequality follows because of the validity of the Triangle Inequality with the absolute value on the real line. What is the following set? U=fx= (x1;x2)2R2jd0(x;O) = 1g: We can dene an analogous metric, called the taxicab metric, on Rn. d0(x;y) =nX i=1jxiyij: c 1999, David Royster Introduction to Topology For Classroom Use Only 12 CHAPTER 2. METRIC SPACES Example 2.1.5 [The Max Metric on Rn] Another metric for Rnis given by taking the largest of the dierences of the coordinates of xandy. d00(x;y) = maxfjxiyijgn i=1: Example 2.1.6 [The Discrete Metric] For any set X, dene d(x;y) =( 0 ifx=y 1 ifx6=y This denes a metric on X, called the discrete metric . It is usually of little use, except for counterexamples. It does show, though, that every set can be assigned a metric. Example 2.1.7 LetC[a;b] denote the set of all continuous real-valued functions dened on the interval [ a;b]. Forf;g2C[a;b] dene (f;g) =Zb ajf(x)g(x)jdx: The fact that is a metric follows from the usual properties of the Riemann integral. This metric measures the distance between two functions to be the area between the two graphs from x=atox=b. Example 2.1.8 For the set C[a;b] dene0by 0(f;g) = lubfjf(x)g(x)jjx2[a;b]g: The metric is called the supremum metric or the uniform metric forC[a;b]. It measures the distance between fandgto be the supremum of the vertical distances from points ( x;f(x)) to (x;g(x)) on the graphs of fandgon the closed interval [ a;b]. Denition 2.2 A numberuis an upper bound for a setAof real numbers provided thataufor alla2A. If there is a smallest upper bound u0forA, that is an upper bound that is less than or equal to all other upper bounds for A, thenu0is called theleast upper bound or supremum ofA. The least upper bound for a set Ais denoted by lubAorsupA. Denition 2.3 A number`is an lower bound for a setAof real numbers provided that`afor alla2A. If there is a largest lower bound `0forA, that is a lower bound that is less than or equal to all other lower bounds for A, then`0is called the greatest lower bound or inmum ofA. The greatest lower bound for a set Ais denoted by glbAorinfA. c 1999, David Royster Introduction to Topology For Classroom Use Only 2.1. DEFINITION AND SOME EXAMPLES 13 A very basic property of the real numbers is included in the following two state- ments: The Least Upper Bound Property : Every non-empty set of real numbers which has an upper bound has a least upper bound. The Greatest Lower Bound Property : Every non-empty set of real numbers which has a lower bound has a greatest lower bound. We will accept the rst property as an axiom of the real number system. The second property follows from the rst. Denition 2.4 Let(X;d)be a metric space and let Abe a non-empty subset of X. Iffd(x;y)kx;y2Aghas an upper bound, then Ais said to be bounded , and lubfd(x;y)kx;y2Agis called the diameter ofA. For completeness, we dene the diameter of the empty set to be 0. If the set Xis bounded, then we call (X;d)a bounded metric space. Ifx2X, then the distance fromxtoAis dened by d(x;A) = glbfd(x;y)jy2Ag: c 1999, David Royster Introduction to Topology For Classroom Use Only