General Topology

By Stephen Willard

Among the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Its treatment encompasses two broad areas of topology: "continuous topology," represented by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and "geometric topology," covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory. Many standard spaces are introduced in the related problems that accompany each section (340 exercises in all). The text's value as a reference work is enhanced by a collection of historical notes, a bibliography, and index. 1970 edition. 27 figures.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 12, 2012
• ISBN: 9780486131788

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Copyright

Copyright © 1970 by Addison-Wesley Publishing Company Copyright © Renewed 1998 by Stephen Willard All rights reserved.

Bibliographical Note

This Dover edition, first published in 2004, is an unabridged republication of the work originally published by the Addison-Wesley Publishing Company, Reading, Massachusetts, in 1970.

International Standard Book Number: 0-486-43479-6

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501

9780486131788

Preface

This book is designed to develop the fundamental concepts of general topology which are the basic tools of working mathematicians in a variety of fields. The material here is sufficient for a variety of one- or two-semester courses, and presupposes a student who has successfully mastered the material of a rigorous course in advanced calculus or real analysis. Thus it is addressed primarily to the beginning graduate student and the good undergraduate.

A principal goal here has been to seek some sort of balance, in the treatment, between two broad areas into which general topology might (rather arbitrarily and, of course, inaccurately) be divided. The first, which could be called continuous topology, centers on the results about compactness and metrization which are the indispensable tools of the modern analyst. This is what Kelley has labeled what every young analyst should know, and is represented here by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces and function spaces. The second area, which might be called geometric topology, is primarily concerned with the connectivity properties of topological spaces and provides the cores of results from general topology which are necessary preparation for later courses in geometry and algebraic topology. This core is formed here by a series of nine sections on connectivity properties, topological characterization theorems and homotopy theory. By suitable surgical intervention, mixed audiences can be taught a mixture of the two approaches, using whatever recipe the instructor likes best. To aid in the concoction of such recipes this preface is followed by a table of some of the important topics in the book together with a list of the material which is prerequisite for each.

While trying to maintain the balance just described, I have also tried to keep in mind the potential uses of such a book both as a text and as a reference source. Thus, in a concession to pedagogy, I have paced the book rather more slowly at the beginning than at the end and have concentrated motivational comments at the beginning. I have also attempted to keep the pedagogical lines of force transparent by paring the material of each section down to what I believe is fundamental. At the same time, I have included a large selection of exercises (over 340, each containing several parts), which provide drill in the techniques developed in the text, develop limiting counterexamples and provide extensions of, and parallels to, the theory presented in the text. Some of the theoretical exercises are suitable for extended development and discussion in the classroom, and all should enhance the value of the book as a reference source. Worth particular mention are the exercises on normed linear spaces and topological groups, and many of the exercises in the sections on compactness, compactification, metrization and the Stone-Weierstrass theorem. To facilitate its use as a reference source, I have included at the end of the book a collection of background notes for each section, a large (but certainly not exhaustive) bibliography and an index as comprehensive as my patience would allow.

The primary organization of the book is into forty-four sections; chapter headings are provided, but not as a referencing device; they serve only to collect the sections into coherent groups. Within each section, the definitions, examples and theorems are further numbered consecutively so that Theorem 25.3 appears as the third item (not necessarily the third theorem) in Section 25. The one exception to this rule is Section 1, on set theory, where the material is somewhat condensed and the numbers 1.1, 1.2, . . . serve to designate subsections rather than specific results. One note of caution seems advisable. A reference to a theorem number only, omitting the word theorem, should serve as a warning that the relevant observation may be made in the remarks following the proof of the theorem, rather than in the statement of the theorem itself. (This happens infrequently, however, and most references, even of this type, are to the numbered theorem itself.) Each section ends with a set of exercises, lettered consecutively; most exercises consist of several parts. A reference to 3E is a reference to the fifth exercise in Section 3; where more precision is needed, 3E.3 is used to designate the third part of this exercise.

A few notational and terminological conventions deserve special mention. Following the lead of Halmos and Kelley, we replace the cumbersome if and only if by iff and denote the end of a proof by ■. When discussing statements of the form "P iff Q, we occasionally use necessity to mean if P then Q and sufficiency to mean P if Q". Square brackets are used nonmathematically in two contexts in this book. At the end of an exercise, they enclose hints to the solution of that exercise, and placed at the end of an item in the bibliography, they enclose a reference to the review of that item in the Mathematical Reviews or (for items written between 1930 and 1940) the Zentralblatt.

Anyone who writes a book of this sort accumulates a sea of outstanding debts. My own personal sea has been fed by more rivers of kindness than I can count; many have no doubt achieved the status of underground streams and been forgotten. The one I cannot forget created the sea long before this project was conceived, and I here acknowledge my greatest debt to A. H. Stone. J’en suis pas digne.

The presentation here has been affected by countless conversations with friends and colleagues, who were not always aware they were speaking for posterity. I apologize, mentioning particularly Donald Plank, Melvin Henriksen, W. W. Comfort, Don Johnson, Ta Sun Wu, John Isbell, Anthony Hager and Phillip Nanzetta. A great many students deserve my thanks for stoically suffering through earlier versions of the manuscript: These include my own at Lehigh, Case Western Reserve and the University of Alberta, as well as those of Professor Johnson at New Mexico State University and Professor Comfort at Wesleyan University. Especially, parts of the manuscript were assiduously edited by Robert Shurtleff, and critically reviewed by the students in Professor Comfort’s class. They will, I think, recognize their influence in the ultimate presentation.

If I mention the students who have suffered through one or another of the early versions of this manuscript, I cannot neglect my wife, Mary, who has suffered through every version, both as wife and as proof reader.

The typing was done by Elizabeth Roach and Rosemary Pappano. Virtually every mistake that survived their typing was my own and I am shaken to report that they caught several of my best and most subtle errors, mathematical and otherwise.

Case Western Reserve University deserves my thanks for making it possible for me to avoid dividing my time and myself between the classroom and preparation of this manuscript in the fall of 1968. Parts of the manuscript were prepared during my tenures on several grants from the National Science Foundation.

Edmonton, Alberta

S.W.

April 1970

For completeness, I will list also some of the special sections and their dependence on previous material.

Table of Contents

Title Page

Copyright Page

Preface

Chapter 1 - Set Theory and Metric Spaces

Chapter 2 - Topological Spaces

Chapter 3 - New Spaces from Old

Chapter 4 - Convergence

Chapter 5 - Separation and Countability

Chapter 6 - Compactness

Chapter 7 - Metrizable Spaces

Chapter 8 - Connectedness

Chapter 9 - Uniform Spaces

Chapter 10 - Function Spaces

Historical Notes

Bibliography

Index

Table of Contents

Title Page

Copyright Page

Preface

Chapter 1 - Set Theory and Metric Spaces

Chapter 2 - Topological Spaces

Chapter 3 - New Spaces from Old

Chapter 4 - Convergence

Chapter 5 - Separation and Countability

Chapter 6 - Compactness

Chapter 7 - Metrizable Spaces

Chapter 8 - Connectedness

Chapter 9 - Uniform Spaces

Chapter 10 - Function Spaces

Historical Notes

Bibliography

Index

Chapter 1

Set Theory and Metric Spaces

1 Set theory

The material of this section is introduced primarily to serve as a review for those with some background in set theory and as an introduction to our notational conventions and terminology. The reader entirely unfamiliar with any aspect of set theory should not be content with the intuitive discussion given here, but should consult one of the standard references on the subject (see the notes).

Most of the material in this book is accessible to anyone who understands 1.1 through 1.8 below. It is recommended that the remainder of this section be skipped on first reading and referred to later as needed.

1.1 Sets. A set, family or collection is an aggregate of things (for example, numbers or functions or desks or people), called the elements or points of the set. If a is an element of the set A we write a ∈ A and if this is false we write a ∉ A.

If A is a set and S is a statement which applies to some of the elements of A, the set of elements a of A for which S(a) is true is denoted {a ∈ A ∣ S(a)}. Thus if N is the set of positive integers, the positive divisors of 6 form the set {a ∈ N ∣ab = 6 for some b ∈ N}. In the case of small sets, such as this one, it is easy to describe the set by listing its elements in brackets. Thus the set just given is the set {1, 2, 3, 6}.

This discussion is rather naïve and leads to certain difficulties. Thus if P is the set of all sets, we can apparently form the set Q = {A ∈ P ∣A ∉ A}, leading to the contradictory Q ∈ Q iff Q ∉ Q. This is Russell’s paradox (see Exercise 1A) and can be avoided (in our naïve discussion) by agreeing that no aggregate shall be a set which would be an element of itself.

1.2 Elementary set calculus. If A and B are sets and every element of A is an element of B, we write A ⊂ B or B ⊃ A and say A is a subset of B or B contains A. The collection P(A) of all subsets of a given set A is itself a set, called the power set of A.

We say sets A and B are equal, A = B, when both A ⊂ B and B ⊂ A. Evidently, A and B are equal iff they have the same elements.

We write A − B to denote the set {a ∈ A∣a ∉ B} and (unlike some writers) use this notation even when B is not a subset of A, i.e., even when B ⊄ A. When we do have B ⊂ A, A − B is called the complement of B in A.

The empty set, ⌀; is the set having no elements. By the criterion for equality of sets, there is only one empty set and, by the criterion for containment, it is a subset of every other set.

Note that element and subset are different ideas. Thus, for example, x ∈ A iff {x} ⊂ A.

A few sets will keep recurring and we will establish now a conventional notation for them.

R: the set of real numbers,

Rn: Euclidean n-space,

N: the set of positive integers,

I: the closed interval [0, 1] in R,

Q: the set of rational numbers in R,

P: the set of irrational numbers in R,

Sn: the n-sphere, {x ∈ Rn+1 ∣|x| = 1}.

Eventually, each of these sets will be assumed to carry some usual structure (a metric, topology, uniformity or proximity) unless the contrary is noted. Additional less often used conventional notations will be introduced in the text. All can be found in the index.

1.3 Union and intersection. If Λ is a set and, for each λ ∈ Λ, Aλ is a set, the union of the sets Aλ∈Λ Aλ, of all elements which belong to at least one Aλ. When no confusion about the indexing can result, we will write the union of the sets AAλ. The intersection of the sets Aλ∈Λ AAλ, of all elements which belong to every Ais the collection {Aλ∣ λ ∈ Λ}, the union and intersection of the sets A, respectively.

When only finitely many sets A1, . . . , An are involved, the alternative notations A1 ∪⋯∪An Ak are sometimes used for the union of the Ak, while A1 ∩⋯∩An Ak sometimes denotes their intersection. When denumerably many sets A1, A2, . . . are involved, their union will sometimes be denoted by A1 ∪ A, their intersection by A1 ∩ A

We say A meets B iff A ∩ B ≠ . ø Otherwise, A and B are disjoint. of sets is pairwise disjoint iff whenever A, B , A ∩ B = ø.

For those who wish to test themselves on the concepts just introduced, here are a few easily proved facts:

A ⊂ B iff A ∪ B = B,

A ⊂ B iff A ∪ B = A,

is the empty collection of subsets of A.

A ∪ B = A ∪ (B − A).

A ∩ (B ∪ C) = (A ∩ B) ∪ C iff C ⊂ A.

1.4 Theorem. If A is a set, Bλ ⊂ A for each λ ∈ Λ and B ⊂ A, then

Proof. a) If x ∈ A Bλ), then x ∈ A and x ∉ Bλ for any λ, so x ∈ A − Bλ for each λ; hence x (A − Bλ). Conversely if x (A − Bλ), then for each λ, x ∈ A and x ∉ Bλ; hence x ∈ A Bλ. Thus x ∈ A Bλ iff x (A − Bλ), so that

.

b) Similar to (a). See Exercise 1B.

c) If x ∈ B Bλ, then x ∈ B Bλ; thus x ∈ B and x for some λ0. Hence x (B ∩ Bλ(B ∩ Bλ), then x ∈ B for some λ0 ∈ Λ; thus x ∈ B and x , so that x ∈ B and x Bλ. Hence x Bλ. We have shown x ∈ B Bλ) iff x (B ∩ Bλ.

d) Similar to (c). See Exercise 1B. ■

1.5 Small Cartesian products. If x1 and y2 are distinct elements of some set, the two-element sets {x1, x2} and {x2, x1} are, by the criterion for set equality, the same. It is useful to have a device for reflecting priority as well as membership in this case, and it is provided by the notion of the ordered pair (x1, x2). By definition, ordered pairs (x1, x2) and (y1, y2) are equal iff x1 = y1 and x2 = y2. For a somewhat more formal approach to ordered pairs, see Exercise 1C.

Now if X1 and X2 are sets, the Cartesian product X1 × X2 of X1, and X2 is defined to be the set of all ordered pairs (x1, x2) such that x1 ∈ X1 and x2 ∈ X2. This definition, for example, gives the plane as the set of all ordered pairs of real numbers. Other examples: S¹ × I is a cylinder, S¹ × S¹ is a torus, R × Rn = Rn+1 .

Once defined for two sets, Cartesian products of any finite number of sets can be defined by induction; thus, the last example in the previous paragraph could be taken as the definition of Rn+1.

For more about finite Cartesian products, and for a bridge between the definition given here and the definition provided in Section 8 for products of infinitely many sets, see Exercise 1D.

1.6 Functions. A function (or map) f from a set A to a set B, written f : A → B, is a subset of A × B with the properties:

For each a ∈ A, there is some b ∈ B such that (a, b) ∈ f.

If (a, b) ∈ f and (a, c) ∈ f, then b = c.

More informally, we are requiring that each a ∈ A be paired with exactly one b ∈ B. The relationship (a, b) ∈ f is customarily written b = f(a) and functions are usually described by giving a rule for finding f(a) if a is known (rather than, for example, by giving some geometric or other description of the subset f of A × B). This reflects the common point of view, which is prone to regard a function not so much as a static subset of A × B as a black box which takes in elements of A and spits out elements of B.

When regarded as a set in its own right, the collection of functions from A to B is denoted BA.

If f : A → B and C ⊂ A, we define f(C) = {b ∈ B∣ b = f(a) for some a ∈ A}. If D ⊂ B, we define f-1(D) = {a ∈ A ∣f(a) ∈ D}. Hence every function f: A → B induces functions ƒ: P(A) → P(B) and f-1: P(B) → P(A). (and here we are following the unfortunate, but common, practice of denoting the elevation of f from A to P(A) by f also). The properties of these induced functions are investigated in Exercise 1H, which should be mandatory for anyone who cannot provide easily the answers to the questions it poses.

Note that if f : A → B, then f-1(B) = A but it need not be true that f(A) = B. We call f(A) the image of f (or the image of A under f), calling B the range of f and A the domain of f. When f(A) = B, we say f is onto B. Note also that, for b ∈ B, f-1({b}) [which is always abbreviated ƒ-1(b)] may consist of more than one point; in extreme cases, we may have f-1(b) = A. When such behavior is proscribed, f is called a one-one function. In addition to the usual requirements for a function, then, a one-one function f : A → B must evidently obey the rule: a1 ≠ a2 ⇒ ƒ(a1) ≠ f(a2). In words, such a function takes distinct elements of A to distinct elements of B.

If f : A → B and g : B → C, then f and g determine together a natural function, their composition g ∘f: A → C, defined by

(g ∘ f)(a) = g[f(a)], for a ∈ A.

More formally, (a, c) ∈ g∘f iff for some b ∈ B, (a, b) ∈ f and (b, c) ∈ g. Less formally, put two black boxes end to end.

1.7 Special functions. A function f : N → A is called a sequence in A. It can be described by giving an indexed list x1, x2, . . . of its values at 1, 2, . . . and this is often abbreviated (xn)n∈N or even simply (xn). Thus ƒ(n) = 1/n, (1/n)n∈N and 1, 1/2....., 1/n, . . . describe the same sequence in R.

A real-valued function on A is a function on A whose range is R. The collection RA of all real-valued functions on A inherits an algebraic structure from R since we can define addition, multiplication and scalar multiplication in RA as follows: given a ∈ A and r ∈ R,

(f + g)(a) = f(a) + g(a),

(fg)(a) = f(a)g(a),

(rf)(a) = r[f(a)].

For this and other reasons, the real-valued functions merit special attention in any branch of mathematics, and topology is no exception.

The identity function on any set A is the function i: A → A defined by i(a) = a for each a ∈ A. More generally, if B ⊂ A, the inclusion j: B → A is the function j(b) = b for each b ∈ B.

1.8 Relations. A relation R on a set A is any subset of A × A. (Thus every function from A to A is a relation on A, but not all relations on A have the properties required of functions.) If R is a relation on A, we usually denote the relationship (a, b) ∈ R by aRb. For example, {(n1, n2) ∈ N × N ∣n1 < n2} is a relation on N and it would be typical to denote this relation by <, so that (n1, n2) ∈ < iff n1 < n2.

A relation R on A is called reflexive iff aRa for each a ∈ A, symmetric iff aRb implies bRa for all a, b ∈ A, antisymmetric iff aRb and bRa implies a = b for all a, b ∈ A and transitive iff aRb and bRc implies aRc for all a, b, c ∈ A. For example, < is a transitive relation on R, ≤ is a reflexive, antisymmetric, transitive relation on R, ≠ is a symmetric relation on R.

An equivalence relation on A is a reflexive, symmetric and transitive relation on A. As an example, let f be any function from A to B and define a relation R on A by xRy iff f(x) = f(y). For other examples, see Exercise 1E.

If R is an equivalence relation on A, the equivalence class (or R-equivalence class where confusion is possible) of a ∈ A is the set [a] = {a′ ∈ A |a′Ra}. If a, b ∈ A, note that either [a] = [b] (and this happens precisely when aRb) or else [a] ∩ [b] = ø. Since a ∈ [a] for each a ∈ A, the sets [a], for a ∈ A, evidently form a partition of A, i.e., they are disjoint sets whose union is A. For example, if R is the equivalence relation introduced in the preceding paragraph, the equivalence class of a ∈ A is the set f−1[f(a)]. Other examples can be found in 1E.

1.9 Order relations. A relation R on A is a partial order provided R is reflexive, antisymmetric and transitive. Thus ≤ is a partial order on R. It is the model partial order and thus it is customary to denote any partial order on any set by ≤. In this context, ≥ is defined by a ≥ b iff b ≤ a.

Associated with any partial order ≤ on A is a relation < defined by a < b iff a ≤ b and a ≠ b. Note that < is not reflexive or symmetric, but it is transitive and has the property that for any a and b in A, if a < b, then b ≮ a. A transitive relation with this property will be called a strict order. Thus every partial order determines a strict order. Conversely, any strict order < determines a partial order ≤ defined by a ≤ b iff a < b or a = b. Moreover the passage from a partial order ≤ to its associated strict order < to the partial order determined by < returns us to ≤, and the assertion remains true with strict order and partial order interchanged. Thus, in dealing with a partially ordered set, the symbol < has a well-defined meaning.

A set A is linearly ordered by a partial order ≤ provided that for any a, b ∈ A exactly one of a < b, b < a or a = b holds. Then ≤ is called a linear order.

If ≤ is a partial order on A, the smallest element of A, if it exists, is the element a0 such that a0 ≤ a for each a ∈ A, and the largest element of A, if it exists, is the element a1, such that a ≤ a1, for each a ∈ A. Smallest (largest) elements are unique, when they exist, by antisymmetry. They may not exist: R with the order ≤ has no smallest or largest element.

A set A is well-ordered if it has a linear order ≤ such that every subset of A has a smallest element (in the linear order induced on that subset by the linear order on A). The set N of positive integers is well-ordered by its usual order, the real line R is not.

1.10 Minimal and maximal elements. If A is partially ordered by ≤, an element b0 of A is a minimal element of A provided b ≤ b0 implies b = b0 for each b ∈ A, and b1 is a maximal element of A provided b1 ≤ b implies b1 = b for each b ∈ A. If a smallest (largest) element exists in A, then it is the unique minimal (maximal) element of A. In Fig. 1.1, where x < y is represented by a rising line connecting x to y, we find an example of a set with a unique maximal element b which is not a largest element, so the converse fails.

Figure 1.1

The reader is invited to draw a diagram illustrating that maximal elements need not be unique.

The least upper bound (lub) of a subset B of a partially ordered set A is the smallest element of the set {a ∈ A∣ b ≤ a for each b ∈ B}. It may or may not exist and, when it does, it may or may not belong to B. When it exists, it is unique. The greatest lower bound (glb) of B is similarly defined.

1.11 Lattices. A partially ordered set L is a lattice iff each two-element set {a, b} in L has a least upper bound a ∨ b and a greatest lower bound a ∧ b. If every nonempty subset of L has a least upper bound and a greatest lower bound, L is a complete lattice. Lattices having a least element 0 and a greatest element 1 are called complemented iff for each a ∈ L, there is some a′ ∈ L such that a ∨ a′ = 1, a ∧ a′ = 0. A lattice is distributive iff for all a, b, c ∈ L,

a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

and

a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).

These rules are redundant since either can be deduced from the other.

A Boolean lattice is a lattice with 0 and 1 which is complemented and distributive.

The model lattice for most purposes is the set P(A) of all subsets of a fixed set A. This becomes a complete Boolean lattice when partially ordered by the relation B ≤ C iff B ⊂ C. (See Exercise 1K.)

1.12 Cardinality. If A and B are sets, we say A is equipotent with B iff there is a one–one function f from A onto B. Intuitively, equipotent sets have the same number of elements. We now postulate the existence of sets, called cardinal numbers, so chosen that every set A is equipotent with precisely one cardinal number, called the cardinal number of A and denoted ∣A∣.

If C and D are cardinal numbers, we say C ≤ D iff there is a one–one function f : C → D. The result is a partial order on any family of cardinal numbers. Let us see what this says:

a) ≤ is reflexive: given a cardinal number C, there is a one–one function f: C → C. The identity function will do nicely.

b) ≤ is antisymmetric : given cardinal numbers C and D, if one–one functions f : C → D and g: D → C can be found, then C = D. This is the Cantor–Bernstein theorem, which in more general form says that if one–one functions f: A → B and g: B → A can be found, then there is a one-one function carrying A onto B. (Existence of a one–one, onto function between cardinal numbers C and D ensures that C = D. Why?) A proof of the Cantor–Bernstein theorem is given in Exercise 1J.

c) ≤ is transitive: given cardinal numbers C, D and E and one–one functions f: C → D and g: D → E, there is a one-one function h: C → E. Here, the composition g ∘ ƒ : C → E will serve.

In fact, any set of cardinal numbers is well-ordered by the relation ≤, although we will not prove this, deferring to any of the standard references on set theory (see the notes).

Recalling that ∣A∣ denotes the cardinal number of A, evidently

i) ∣A∣= ∣B∣ iff A and B are equipotent,

ii) ∣A∣ ≤ ∣B∣ iff A is equipotent with some subset of B.

1.13 Special cardinals. We will distinguish notation for certain cardinal numbers. The empty set is the cardinal 0, and the cardinal number n is the set {0, . . . , n - 1}. A set A is denumerable iff A is equipotent with N and, in this case, we write ∣A∣ = ℵ0. A set A is said to have the cardinal of the continuum, iff A is equipotent with R, and then we write ∣A∣ = c. A set A is countable iff it is denumerable or has cardinal number n for some n = 0, 1, 2, . . . ; otherwise, A is uncountable. The elements of a countable set A can be listed in a (finite or infinite) sequence a1, a2, . . . and such a listing is called an enumeration of the elements of A.

1.14 Facts about countability. a) n < ℵ0 < c,

b) The union of countably many countable sets is countable,

c) The product of two countable sets is countable,

d) The set Q of rational numbers is countable.

Proof a) It is clear that n < ℵ0 and, since N is equipotent with the subset {1/n | n = 1, 2, . . .} of R, that ℵ0 ≤ c. To show ℵ0 ≠ c, it is enough to show that there is no one-one function from N onto I. If such a function f: N → I exists, let the decimal expansion of f(n···. Define .b1b2 . . . by taking bk to be 5 if akk ≠ 5, bk to be 7 if akk = 5. Then .b1b2 · · · is an element of I which can appear nowhere among the values of f, since it differs from f(n) in the nth place, for each n = 1, 2, . . . . This contradicts the assumption that f is onto, showing no such function can exist, and completes the proof of (a).

b) Let {A1, A2, . . } be a countable collection of countable sets. Set B1 = A1 and, for n > 1, Bn = An Ak. Then each Bn is countable and

Enumerate the elements of each Bn as follows:

and define fBn by f(1) = b11, f(2) = b21, f(3) = b12, f(4) = b13, f(5) = b22, f(6) = b31, . . . and so on, following the scheme indicated by the arrows. The result is a one-one function f from N , and the proof is complete.

c) If A and B are countable, enumerate the elements of B as b1, b2, . . . and let An = A × {bn} = {(a, bn) | a ∈ A}. Then An is countable for each n = 1, 2, . . . and A × B An; thus A × B is countable by part (b).

d) Write each element of Q in the form m/n, where m and n are integers in lowest terms. Then the function defined by f(m/n) = (m, n) maps Q in one–one fashion onto a subset of N × N. Since N × N is countable by part c), Q is countable. ■

1.15 Cardinality and the power set. It is possible to develop an arithmetic of cardinal numbers. We limit ourselves here to the definition of exponentiation. If A and B are sets, |A||B| is defined to be |AB| (recall AB denotes the set of all functions f: B → A). The reader will verify in Exercise 1I that this definition gives the right answer if |A| = n and |B| = m, where n and m are integers.

Let us pay particular attention to the cardinal number 2|A| where A is a set. Now 2 = {0, 1} and hence 2|A| = |2A| = |{0, 1}A| is the cardinal number of the set of functions f: A → {0, 1}. Such a function f determines and is completely determined by the subset B = {a ∈ A | f(a) = 1} of A (f is called the characteristic function of B) and hence 2|A| = |P(A)|.

By writing elements of I < 2c. It is generally true for any cardinal number α that α < 2α; put another way, for any set A, |A| < |P(A)|. This is Cantor s theorem (Exercise 1I).

1.16 The continuum hypothesis. The continuum hypothesis states that there are no sets A for which ℵ0 < |A. It has been proved independent of the other axioms needed to develop set theory (see notes); that is, either it or its negation can consistently be added to the other axioms. At present, intuition has provided us with little basis for preferring one assumption over the other (although in most contexts in which it arises, it, rather than its negation, is assumed) and it is definitely in order to attempt to eliminate from any proof any use of the continuum hypothesis. It follows, in the same vein, that whenever it or its negation is assumed, this should be explicitly pointed out.

1.17 The axiom of choice. The following axiom is assumed by most mathematicians when they need it, to the unremitting disgust of a few. We give it in two equivalent forms:

Axiom of choice

If {Aλ| λ ∈ Λ} is a family of nonempty pairwise-disjoint sets, there is a set B Aλ such that B ∩ Aλ has exactly one element, for each λ ∈ Λ.

If {Aλ | λ ∈ Λ} is an indexed family of nonempty pairwise-disjoint sets, there is a function fAλ such that f(λ) ∈ Aλ, for each λ ∈ Λ (f is called a choice function).

It is left to the reader to decide that these two statements both say the same thing. What they say is: given any collection of sets, however large, we can pick one element from each set in the collection. It bothers some people because it asserts the existence of a set (i.e., B in part (a)) without giving enough information to determine that set uniquely (by applying a finite number of rules), and it is the only formal set-theoretic axiom which does this. For this reason it is customary to mention the axiom of choice whenever it is used. It need not be used if the number of sets is finite. In particular, if A is a nonempty set, the statement "choose a ∈ A" need not be supported by an appeal to the axiom of choice.

The status of the axiom of choice bears some resemblance to that of the continuum hypothesis, with some differences. It, too, is known to be independent of the other axioms of set theory (that is, it or its negation can be consistently assumed), but it enjoys the status of an accepted part of the theory of sets in the minds of most modern mathematicians; that is, the intuition of almost all mathematicians now is that the axiom of choice should be assumed where needed without hesitation. Moreover, it is usually much clearer that, where it is used, it is needed, so that its presence does not usually provoke the same frenzy of attempt to eliminate it.

1.18 Alternative forms of the axiom of choice. We now provide some alternative, often-used forms of the axiom of choice. We say a family of sets is of finite character iff each finite subset of a member of the family is also a member, and each set belongs if each of its finite subsets belong.

Theorem. The following statements are all equivalent:

a) (Axiom of choice): If {Aλ | λ ∈ Λ} is an indexed family of nonempty pairwise disjoint sets, there is a set B Aλ such that B ∩ Aλ is exactly one element for each λ ∈ Λ.

b) (Zorn’s Lemma): If each chain (linearly ordered set) in a nonempty partially ordered set A has an upper bound, then A has a maximal element.

c) (Zermelo’s Theorem): Every set can be well-ordered.

d) (Tukey’s Lemma): Each nonempty family of sets of finite character has a maximal element.

As with the axiom of choice, it is customary to mention any one of these wherever it is used. The proof of equivalence will not be given here; it can be found in any standard reference.

1.19 Ordinals. For our purposes, it will be sufficient to postulate the existence of an uncountable well-ordered set Ω with a largest element ω1, having the property that if α ∈ Ω with α < ω1, then {β ∈ Ω | β ≤ α} is countable. Such a set Ω exists if there exists any uncountable well-ordered set; see Exercise 1L. The elements of Ω are ordinals with ω1 being the first uncountable ordinal and Ω0 = Ω–{ω1} being the set of countable ordinals.

If α and β are ordinals with α < β, we say α is a predecessor of β and β is a successor of α. We call α an immediate predecessor of β, and β an immediate successor of α, if β is the smallest ordinal larger than α. Every ordinal α has an immediate successor, often denoted α + 1; some ordinals, called the limit ordinals have predecessors without having an immediate predecessor (ω1, for example). The others are nonlimit ordinals.

To build a picture of Ω, observe that it has a least element, which we denote 1 for now. The immediate successor of 1 will be denoted 2, the immediate successor of 2 will be denoted 3, and so on, so that we can regard the first few elements of Ω as being the positive integers 1, 2, 3, . . . . Since Ω0 is well-ordered, there is a smallest ordinal larger than all of 1, 2, 3, . . . . It is called the first infinite ordinal ω0. It is still only a countable ordinal; it and its first few successors ω0 + 1, ω0 + 2, . . . evidently form another copy of N tacked on behind the first. The smallest ordinal larger than these is denoted 2ω0, and we can apparently continue in this fashion through 3ω0, 4ω0, . . . by adding denumerably many copies of N one after the other.

1, 2, 3, . . . , ω0, ω0 + 1, ω0 + 2, . . . , 2ω0, 2ω0 + 1, 2ω0 + 2, . . . , . . . .

, . . . . The smallest ordinal larger than all these is still countable however, so the process continues. In fact, ω1 is unreachable by countable operations such as this, by the next theorem.

1.20 Theorem. If A is a countable subset of Ω not containing ω1, then lub A < ω1.

Proof. For each α ∈ A, {β ∈ Ω |β ≤ α} is countable. Since A is countable, the union of these sets, namely B = {β ∈ Ω | β ≤ α for some α ∈ A}, is also countable. Let γ be the smallest element of Ω not contained in B. Then β ∈ B iff β < γ, so γ has a countable number of predecessors, and hence γ < ω1. But γ is an upper bound for A, so lub A < ω1. ■

1.21 Induction. The following theorem is a statement of the principle of mathematical induction. To prove it, we accept as obvious the fact that the positive integers N form a well-ordered set.

Theorem. Let S(n) be a statement which is true or false, for n = 1, 2, . . . . If

S(1) is true,

S(n) is true implies S(n + 1) is true, for n = 1, 2, . . . ,

then S(n) is true for all n.

Proof. If the set F of all integers n for which S(n) is false is nonempty, then it has a least element n, and n ≠ 1 by (a). Since n > 1, n − 1 ∈ N, and n − 1 ∉ F, so S(n − 1) is true. But then S(n) is true, by (b) ; this contradiction establishes that F = ø. ■

As an example, we prove that 1 + 2 +⋯+ n = n(n + 1)/2 for any positive integer n. The formula certainly works for n = 1. Suppose it works for n. Then

which is the form the formula should take for n + 1. The inductive step is now established, so by the principle of mathematical induction, the formula applies to any n.

It is also instructive to point out an often used incorrect form of application of the principle of mathematical induction. A typical (wrong) argument would sound like this: "{1} is a finite set, and, if {1, . . . , n} is a finite set, so is {1, . . . , n + 1}. Therefore the positive integers form a finite set." This argument looks as absurd as it is, but uses of the principle of mathematical induction just as ridiculous logically are often submitted by those new to it.

1.22 Transfinite induction. A second method of induction, the principle of transfinite induction, can be applied to statements indexed by a well-ordered set of any sort. We will not need it in any form other than as stated here, however:

Theorem. For each ordinal α ∈ Ω0, let S(α) be a statement which is true or false. If

S(1) is true,

S(β) is true for all β < α implies S(α) is true,

then S(α) is true for each α ∈ Ω0.

The proof is in no essential way different from the proof of the principle of mathematical induction: one makes the same use of the well-ordering.

Both induction principles can be used as the basis for defining things. For example

f(1) = 1,

f(n + 1) = (n + 1)f(n)

is an inductive definition of the factorial function on N. For an example of definition by transfinite induction, see 31.

1.23 Remarks. The process which topology evolves from, outlined in the next section and the notes, is basic to any pure mathematical discipline. We wish to study a particular property enjoyed by some objects of interest (in this case, continuity of functions on some space) and the efficient way to proceed is to first clean the structure on the space down to the bare bones needed for introducing and developing the property we want. The passage to such abstraction has several well-documented advantages. Among them:

1. Since we have only what is essential, our proofs use only what is essential and thus clarify the nature of the object of study, and the logical dependence of the theorem in question.

2. Proofs become easier. Actually, this is a popular professional myth, with an element of truth. Occasionally, a proof really does get easier as a theorem gets more abstract, but this is offset by the need for more and more interpretive skill on the part of those who would use the theorem. What people really mean when they say proofs become easier is something like this: by establishing some notation and introducing the right definitions and conventions, we can draw together all the theorems about this subject and find common characteristics and even repetitions in their proofs, then prove lemmas which enable us to write large numbers of proofs more succinctly. If the subject matter is carefully chosen, the work done in abstracting the properties needed, establishing notation and proving those lemmas will be more than paid for by the gain in succinctness and clarity of the proofs later on, and by the acquisition of powerful methods for continued investigation of the original objects of study.

Such is the case with topology.

Problems

1A. Russell’s Paradox

The phenomenon to be presented here was first exhibited by Russell in 1901, and consequently is known as Russell’s Paradox.

Suppose we allow as sets things A for which A ∈ A. Let P be the set of all sets. Then P can be divided into two nonempty subsets, P1 = {A ∈ P |A ∉ A} and P2 = {A ∈ P |A ∈ A}. Show that this results in the contradiction: P1 ∈ P1 ⇔ P1 ∉ P1. Does our (naive) restriction on sets given in 1.1 eliminate the contradiction?

1B. De Morgan’s laws and the distributive laws

[see 1.4a), b)].

[see 1.4c), d)].

If Anm is a subset of A for n = 1, 2, . . . and m = 1, 2, . . . is it necessarily true that

1C. Ordered pairs

Show that, if (x1, x2) is defined to be {{x1}, {x1, y2}}. then (x1, x2) = (y1, y2) iff x1 = x2 and y1 = y2.

1D. Cartesian products

1. Provide an inductive definition of "the ordered n-tuple (x1, . . . , xn) of elements x1, . . . , xn of a set" so that (x1; . . . , xn) and (y1, . . . , yn) are equal iff their coordinates are equal in order, i.e., iff x1 = y1, . . . , xn = yn.

2. Given sets X1, . . . , Xn define the Cartesian product X1 × . . . × Xn

by using the definition

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