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    Introduction to Topology: Third Edition
    Introduction to Topology: Third Edition
    Introduction to Topology: Third Edition
    Ebook285 pages4 hours

    Introduction to Topology: Third Edition

    By Bert Mendelson

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    Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. Originally conceived as a text for a one-semester course, it is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems. The book's principal aim is to provide a simple, thorough survey of elementary topics in the study of collections of objects, or sets, that possess a mathematical structure. The author begins with an informal discussion of set theory in Chapter 1, reserving coverage of countability for Chapter 5, where it appears in the context of compactness. In the second chapter Professor Mendelson discusses metric spaces, paying particular attention to various distance functions which may be defined on Euclidean n-space and which lead to the ordinary topology. Chapter 3 takes up the concept of topological space, presenting it as a generalization of the concept of a metric space. Chapters 4 and 5 are devoted to a discussion of the two most important topological properties: connectedness and compactness. Throughout the text, Dr. Mendelson, a former Professor of Mathematics at Smith College, has included many challenging and stimulating exercises to help students develop a solid grasp of the material presented.
    LanguageEnglish
    PublisherDover Publications
    Release dateApr 26, 2012
    ISBN9780486135090
    Introduction to Topology: Third Edition

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    Introduction to Topology - Bert Mendelson

    CHAPTER 1

    Theory of Sets

    1 INTRODUCTION

    As in any other branch of mathematics today, topology consists of the study of collections of objects that possess a mathematical structure. This remark should not be construed as an attempt to define mathematics, especially since the phrase mathematical structure is itself a vague term. We may, however, illustrate this point by an example.

    The set of positive integers or natural numbers is a collection of objects N on which there is defined a function s, called the successor function, satisfying the conditions:

    1. For each object x in N , there is one and only one object y in N such that y = s ( x );

    2. Given objects x and y in S such that s ( x ) = s ( y ), then x = y ;

    3. There is one and only one object in N , denoted by 1, which is not the successor of an object in N , i.e., 1 ≠ s ( x ) for each x in N ;

    4. Given a collection T of objects in N such that 1 is in T and for each x in T , s ( x ) is also in T , then T = N.

    The four conditions enumerated above are referred to as Peano’s axioms for the natural numbers. The fourth condition is called the principle of mathematical induction. One defines addition of natural numbers in such a manner that s(x) = x + 1, for each x in N, which explains the use of the word successor for the function s. What is significant at the moment is the conception of the natural numbers as constituting a certain collection of objects N with an additional mathematical structure, namely the function s.

    We shall describe a topological space in the same terms, that is, a collection of objects together with a specified structure. A topological space is a collection of objects (these objects usually being referred to as points), and a structure that endows this collection of points with some coherence, in the sense that we may speak of nearby points or points that in some sense are close together. This structure can be prescribed by means of a collection of subcollections of points called open sets. As we shall see, the major use of the concept of a topological space is that it provides us with an exact, yet exceedingly general setting for discussions that involve the concept of continuity.

    By now the point should have been made that topology, as well as other branches of mathematics, is concerned with the study of collections of objects with certain prescribed structures. We therefore begin the study of topology by first studying collections of objects, or, as we shall call them, sets.

    2 SETS AND SUBSETS

    We shall assume that the terms object, set, and the relation is a member of are familiar concepts. We shall be concerned with using these concepts in a manner that is in agreement with the ordinary usage of these terms.

    If an object A belongs to a set S we shall write (read, "A in S"). If an object A does not belong to a set S we shall write (read, "A not in S"). If A1, . . . , An are objects, the set consisting of precisely these objects will be written {A1, . . . , An}. For purposes of logical precision it is often necessary to distinguish the set {A}, consisting of precisely one object A, from the object A itself. Thus is a true statement, whereas A = {A} is a false statement. It is also necessary that there be a set that has no members, the so-called null or empty set. The symbol for this set is Ø.

    Let A and B be sets. If for each object , it is true that , we say that A is a subset of B. In this event, we shall also say that A is contained in B, which we write , or that B contains A, which we write .

    In accordance with the definition of subset, a set A is always a subset of itself. It is also true that the empty set is a subset of A. These two subsets, A and Ø, of A are called improper subsets, whereas any other subset is called a proper subset.

    There are certain subsets of the real numbers that are frequently considered in calculus. For each pair of real numbers a, b with a < b, the set of all real numbers x such that is called the closed interval from a to b and is denoted by [a, b]. Similarly, the set of all real numbers x such that a < x < b is called the open interval from a to b and is denoted by (a, b). We thus have , where R is the set of real numbers.

    Two sets are identical if they have precisely the same members. Thus, if A and B are sets, A = B if and only if both and . Frequent use is made of this fact in proving the equality of two sets.

    Sets may themselves be objects belonging to other sets. For example, {{1, 3, 5, 7}, {2, 4, 6}} is a set to which there belong two objects, these two objects being the set of odd positive integers less than 8 and the set of even positive integers less than 8. If A is any set, there is available as objects with which to constitute a new set, the collection of subsets of A. In particular, for each set A, there is a set we denote by 2A whose members are the subsets of A. Thus, for each set A, we have if and only if .

    EXERCISES

    1. Determine whether each of the following statements is true or false:

    (a) For each set .

    (b) For each set .

    (c) For each set .

    (d) For each set .

    (e) For each set .

    (f) There are no members of the set {Ø}.

    (g) Let A and B be sets. If , then .

    (h) There are two distinct objects that belong to the set {Ø, {Ø}}.

    2. Let A , B , C be sets. Prove that if and , then .

    3. Let A 1 , . . . , A n be sets. Prove that if and , then A 1 = A 2 = . . . = A n .

    3 SET OPERATIONS: UNION, INTERSECTION, AND COMPLEMENT

    If x is an object, A a set, and , we shall say that x is an element, member, or point of A. Let A and B be sets. The intersection of the sets A and B is the set whose members are those objects x such that and . The intersection of A and B is denoted by (read, "A intersect B"). The union of the sets A and B is the set whose members are those objects x such that x belongs to at least one of the two sets A, B; that is, either or .* The union of A and B is denoted by (read, "A union B").

    The operations of set union and set intersection may be represented pictorially (by Venn diagrams). In Figure 1, let the elements of the set A be the points in the region shaded by lines running from the lower left-hand part of the page to the upper right-hand part of the page, and let the elements of the set B be the points in the region shaded by lines sloping in the opposite direction. Then the elements of are the points in either shaded region and the elements of are the points in the cross-hatched region.

    Figure 1

    Let . The complement of A in S is the set of elements that belong to S but not to A. The complement of A in S is denoted by Cs(A), S/A, or by S − A. The set S may be fixed throughout a given discussion, in which case the complement of A in S may simply be called the complement of A and be denoted by C(A). C(A) is again a subset of S and one may take its complement. The complement of the complement of A is A; that is, C(C(A)) = A.

    There are many formulas relating the set operations of intersection, union, and complementation. Frequent use is made of the following two formulas.

    THEOREM (DeMorgan’s Laws). Let , . Then

    Proof. Suppose . Then and . Thus, and , or and . Therefore and, consequently,

    Conversely, suppose . Then and and . Thus, and , and therefore . It follows that and, consequently,

    We have thus shown that

    One may prove Formula 3.2 in much the same manner as 3.1 was proved. A shorter proof is obtained if we apply 3.1 to the two subsets C(A) and C(B) of S, thus

    Taking complements again, we have

    EXERCISES

    1. Let , . Prove the following:

    (a) if and only if .

    (b) if and only if .

    (c) if and only if .

    (d) if and only if .

    (e) if and only if .

    (f) if and only if .

    2. Let . Prove the following:

    (a) .

    (b) .

    4 INDEXED FAMILIES OF SETS

    Let I be a set. For each , let Aα be a subset of a given set S. We call I an indexing set and the collection of subsets of S indexed by the elements of I is called an indexed family of subsets of S. We denote this indexed family of subsets of S by (read, "A sub-alpha, alpha in I").

    Let be an indexed family of subsets of a set S. The union of this indexed family, written, , (read "union over α in I of Aα) is the set of all elements such that for at least one index . The intersection of this indexed family, written (read intersection over α in I of Aα") is the set of all elements such that for each . [Note that , for which reason the two occurrences of α in the expression are referred to as dummy indices.]

    As an example, let A1, A2, A3, A4 be respectively the set of freshmen, sophomores, juniors, and seniors in some specified college. Here we have I = {1, 2, 3, 4} as an indexing set, and is the set of undergraduates while .

    If the indexing set I contains precisely two distinct indices, then the union over α in I of Aα is the same as the union of two sets as defined in the previous section; that is,

    Similarly,

    We have allowed for the possibility that the indexing set I is the empty set in which case a careful reading of the definition shows that

    DeMorgan’s laws are applicable to unions and intersections of indexed families of subsets of a set S.

    THEOREM Let be an indexed family of subsets of a set S. Then

    Proof. Suppose . Then ; that is, for each index . Thus for each index and . Therefore,

    Conversely, suppose that . Then for each index . Thus for each index ; that is, . Therefore, and

    This proves 4.1. The proof of 4.2 is left as an exercise.

    Given any collection of subsets of a set S, the concept of indexed family of subsets allows us to define the union or intersection of the aforementioned subsets. We need only construct some convenient indexing set. In the event that the collection of subsets is finite, the finite set {1, 2, . . . , n} of integers is a convenient indexing set. Given subsets A1, A2, . . . , An of S, we shall often write for and, similarly,

    .

    EXERCISES

    1. Let be two indexed families of subsets of a set S. Prove the following:

    (a) For each .

    (b) For each .

    (c)

    .

    (d)

    .

    (e) If for each then

    (f) Let . Then

    2. Let . Then

    3. Let be an indexed family of subsets of a set S. Let . Prove that

    (a) .

    (b) .

    4. Let be an indexed family of subsets of a set S. Let . Prove that

    (a) if and only if for each .

    (b) if and only if for each .

    5. Let I be the set of real numbers that are greater than 0. For each , let A x be the open interval (0, x ). Prove that , . For each , let B x be the closed interval [0, x ]. Prove that .

    5 PRODUCTS OF SETS

    Let x and y be objects. The ordered pair (x, y)* is a sequence of two objects, the first object of the sequence being x and the second object of the sequence being y. Let A and B be sets. The Cartesian product of A and B, written A × B, (read "A cross B") is the set whose elements are all the ordered pairs (x, y) such that and .

    The Cartesian product of two sets is a familiar notion. The coordinate plane of analytic geometry is the Cartesian product of two lines. The possible outcomes of the throw of a pair of dice is the Cartesian product of two sets, A and B, where A = B = {1, 2, 3, 4, 5, 6}. Unless A = B, the two Cartesian products A × B and B × A are distinct.

    A generalization of the Cartesian product of two sets is the direct product of a sequence of sets. Let A1, A2, . . . , An be a finite sequence of sets, indexed by {1, 2, . . . , n}. The direct product of A1, A2, . . . , An, written

    (read "product i equals one to n of Ai") is the set consisting of all sequences (a1, a2, . . . , an) such that . In particular,

    For this reason we shall often write

    for .

    The concept of direct product may be extended to an infinite sequence A1, A2, . . . , An, . . . of sets, indexed by the positive integers. The direct product of A1, A2, . . . , An, . . . , written

    or

    is the set whose elements are all infinite sequences (a1, a2, . . . , an, . . .) such that for each positive integer i.

    The set of points of Euclidean n-space yields an example of a direct product of sets. If for i = 1, 2, . . . , n we have Ai = R, where R is the set of real numbers, then

    is the set of points of a Euclidean n-space. An element is a sequence x = (x1, x2, . . . , xn) of real numbers. In general, if the sets A1, A2, . . . , An are all equal to the same set A, we write

    and call an element an n-tuple.

    EXERCISES

    1. Let . Prove that

    2. Prove that if A has precisely n distinct elements and B has precisely m distinct elements, where m and n are positive integers, then A × B has precisely mn distinct elements.

    3. Let A and B be sets, both of which have at least two distinct members. Prove that there is a subset that is not the Cartesian product of a subset of A with a subset of B. [Thus, not every subset of a Cartesian product is the Cartesian product of a pair of subsets.]

    6 FUNCTIONS

    The most familiar example of a function in mathematics is a correspondence that associates with each real number x a real number f(x). The purpose of marking an examination may be described as the construction of a marking function that makes correspond to each student taking the examination some integer between zero and one hundred. Integration of a continuous function defined on some closed interval [a, b] is another example of a function, namely the correspondence that associates with each object f in this given set of objects the real number

    The concept of function or correspondence need not be restricted to the realm of numerical quantities. The correspondence that associates with each undergraduate in college one of the four adjectives freshman, sophomore, junior, or senior is also an example of a function using correspondence as an undefined concept.

    DEFINITION Let A and B be sets. A correspondence that associates with each element a unique element is called a function from A to B , which we shall write

    or

    DEFINITION Let f : A → B. The subset , which consists of all ordered pairs of the form ( a , f ( a )) is called the graph of f : A → B.

    The graph Γf of a function f:X → Y is the subset of X × Y consisting of precisely those points (x, y) for which the statement f(x) = y is true. This set is sometimes written

    This notation, called the set builder notation, is of the general form {z | P(z) }, where P(z) is some statement which may or may not be true of z. The resulting set is the set of all z, in an appropriate universe, for which P(z) is true.

    Let A and B be sets. Given a subset Γ of A × B there is a function f:A → B such that Γ is the graph of f:A → B if, for each , there is one and only one element of the form .

    (Thus the equivalent definition of a function as a subset with the aforesaid property is frequently employed, in which case for each

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