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基本信息
- 原书名：Topology,Second Edition
- 原出版社： Prentice Hall/Pearson
编辑推荐
本书作者在拓扑学领域享有盛誉
论证严格 条理清晰 实例丰富
内容简介
目录
Preface
A Note to the Reader
Part I GENERAL TOPOLOGY
Chapter 1 Set Theory and Logic
1 Fundamental Concepts
2 Functions
3 Relations
4 The Integers and the Real Numbers
5 Cartesian Products
6 Finite Sets
7 Countable and Uncountable Sets
8 The Principle of Recursive Definition
9 Infinite Sets and the Axiom of Choice
10 Well-Ordered Sets
11 The Maximum Principle
Supplementary Exercises: Well-Ordering
Chapter 2 Topological Spaces and Continuous Functions
12 Topological Spaces
13 Basis for a Topology
14 The Order Topology
A Note to the Reader
Part I GENERAL TOPOLOGY
Chapter 1 Set Theory and Logic
1 Fundamental Concepts
2 Functions
3 Relations
4 The Integers and the Real Numbers
5 Cartesian Products
6 Finite Sets
7 Countable and Uncountable Sets
8 The Principle of Recursive Definition
9 Infinite Sets and the Axiom of Choice
10 Well-Ordered Sets
11 The Maximum Principle
Supplementary Exercises: Well-Ordering
Chapter 2 Topological Spaces and Continuous Functions
12 Topological Spaces
13 Basis for a Topology
14 The Order Topology
前言
This book is intended as a text for a one- or two-semester introduction to topology, at the senior or first-year graduate level.
The subject of topology is of interest in its own right, and it also serves to lay the
foundations for future study in analysis, in geometry, and in algebraic topology. There
is no universal agreement among mathematicians as to what a first course in topology should include; there are many topics that are appropriate to such a course, and not all are equally relevant to these differing purposes. In the choice of material to be treated,I have tried to strike a balance among the various points of view.
Prerequisites. There are no formal subject matter prerequisites for studying most of this book. I do not even assume the reader knows much set theory. Having said that,I must hasten to add that unless the reader has studied a bit of analysis or "rigorous calculus," much of the motivation for the concepts introduced in the first part of the book will be missing. Things will go more smoothly if he or she already has had some experience with continuous functions, open and closed sets, metric spaces, and the like, although none of these is actually assumed. In Part II, we do assume familiarity with the elements of group theory.
Most students in a topology course have, in my experience, some knowledge of the foundations of mathematics. But the amount varies a great deal from one student to another. Therefore, I begin with a fairly thorough chapter on set theory and logic. It starts at an elementary level and works up to a level that might be described as "semi- sophisticated." It treats those topics (and only those) that will be needed later in the book. Most students will already be familiar with the material of the first few sections,but many of them will find their expertise disappearing somewhere about the middle of the chapter. How much time and effort the instructor will need to spend on this chapter will thus depend largely on the mathematical sophistication and experience of the students. Ability to do the exercises fairly readily (and correctly!) should serve as a reasonable criterion for determining whether the student's mastery of set theory is sufficient for the student to begin the study of topology.
Many students (and instructors!) would prefer to skip the foundational material of Chapter 1 and jump right in to the study of topology. One ignores the foundations,however, only at the risk of later confusion and error. What one can do is to treat initially only those sections that are needed at once, postponing the remainder until they are needed. The first seven sections (through countability) are needed throughout the book; I usually assign some of them as reading and lecture on the rest. Sections 9 and 10, on the axiom of choice and well-ordering, are not needed until the discussion of compactness in Chapter 3. Section 11, on the maximum principle, can be postponed even longer; it is needed only for the Tychonoff theorem (Chapter 5) and the theorem
on the fundamental group of a linear graph (Chapter 14).
How the book is organized. This book can be used for a number of different courses.
I have attempted to organize it as flexibly as possible, so as to enable the instructor to
follow his or her own preferences in the matter.
Part I, consisting of the first eight chapters, is devoted to the subject commonly called general topology. The first four chapters deal with the body of material that,in my opinion, should be included in any introductory topology course worthy of the name. This may be considered the "irreducible core" of the subject, treating as it does set theory, topological spaces, connectedness, compactness (through compactness of finite products), and the countability and separation axioms (through the Urysohn metrization theorem). The remaining four chapters of Part I explore additional topics;
they are essentially independent of one another, depending on only the core material of Chapters 1-4. The instructor may take them up in any order he or she chooses.
Part II constitutes an introduction to the subject of Algebraic Topology. It depends on only the core material of Chapters 1-4. This part of the book treats with some thoroughness the notions of fundamental group and covering space, along with their many and varied applications. Some of the chapters of Part II are independent of one another; the dependence among them is expressed in the following diagram:
Certain sections of the book are marked with an asterisk; these sections may be omitted or postponed with no loss of continuity. Certain theorems are marked similarly. Any dependence of later material on these asterisked sections or theorems is indicated at the time, and again when the results are needed. Some of the exercises also depend on earlier asterisked material, but in such cases the dependence is obvious.
Sets of supplementary exercises appear at the ends of several of the chapters. They provide an opportunity for exploration of topics that diverge somewhat from the main thrust of the book; an ambitious student might use one as a basis for an independent paper or research project. Most are fairly self-contained, but the one on topological groups has as a sequel a number of additional exercises on the topic that appear in later sections of the book.
Possible course outlines. Most instructors who use this text for a course in general topology will wish to cover Chapters 1-4, along with the Tychonoff theorem in Chapter 5. Many will cover additional topics as well. Possibilities include the following:
the Stone-Cech compactification (38), metrization theorems (Chapter 6), the Peanocurve (44), Ascoli's theorem (45 and/or 47), and dimension theory (50). I have,in different semesters, followed each of these options.
For a one-semester course in algebraic topology, one can expect to cover most ofPart II.
It is also possible to treat both aspects of topology in a single semester, although with some corresponding loss of depth. One feasible outline for such a course would consist of Chapters 1-3, followed by Chapter 9; the latter does not depend on the material of Chapter 4. (The non-asterisked sections of Chapters 10 and 13 also are independent of Chapter 4.)
The subject of topology is of interest in its own right, and it also serves to lay the
foundations for future study in analysis, in geometry, and in algebraic topology. There
is no universal agreement among mathematicians as to what a first course in topology should include; there are many topics that are appropriate to such a course, and not all are equally relevant to these differing purposes. In the choice of material to be treated,I have tried to strike a balance among the various points of view.
Prerequisites. There are no formal subject matter prerequisites for studying most of this book. I do not even assume the reader knows much set theory. Having said that,I must hasten to add that unless the reader has studied a bit of analysis or "rigorous calculus," much of the motivation for the concepts introduced in the first part of the book will be missing. Things will go more smoothly if he or she already has had some experience with continuous functions, open and closed sets, metric spaces, and the like, although none of these is actually assumed. In Part II, we do assume familiarity with the elements of group theory.
Most students in a topology course have, in my experience, some knowledge of the foundations of mathematics. But the amount varies a great deal from one student to another. Therefore, I begin with a fairly thorough chapter on set theory and logic. It starts at an elementary level and works up to a level that might be described as "semi- sophisticated." It treats those topics (and only those) that will be needed later in the book. Most students will already be familiar with the material of the first few sections,but many of them will find their expertise disappearing somewhere about the middle of the chapter. How much time and effort the instructor will need to spend on this chapter will thus depend largely on the mathematical sophistication and experience of the students. Ability to do the exercises fairly readily (and correctly!) should serve as a reasonable criterion for determining whether the student's mastery of set theory is sufficient for the student to begin the study of topology.
Many students (and instructors!) would prefer to skip the foundational material of Chapter 1 and jump right in to the study of topology. One ignores the foundations,however, only at the risk of later confusion and error. What one can do is to treat initially only those sections that are needed at once, postponing the remainder until they are needed. The first seven sections (through countability) are needed throughout the book; I usually assign some of them as reading and lecture on the rest. Sections 9 and 10, on the axiom of choice and well-ordering, are not needed until the discussion of compactness in Chapter 3. Section 11, on the maximum principle, can be postponed even longer; it is needed only for the Tychonoff theorem (Chapter 5) and the theorem
on the fundamental group of a linear graph (Chapter 14).
How the book is organized. This book can be used for a number of different courses.
I have attempted to organize it as flexibly as possible, so as to enable the instructor to
follow his or her own preferences in the matter.
Part I, consisting of the first eight chapters, is devoted to the subject commonly called general topology. The first four chapters deal with the body of material that,in my opinion, should be included in any introductory topology course worthy of the name. This may be considered the "irreducible core" of the subject, treating as it does set theory, topological spaces, connectedness, compactness (through compactness of finite products), and the countability and separation axioms (through the Urysohn metrization theorem). The remaining four chapters of Part I explore additional topics;
they are essentially independent of one another, depending on only the core material of Chapters 1-4. The instructor may take them up in any order he or she chooses.
Part II constitutes an introduction to the subject of Algebraic Topology. It depends on only the core material of Chapters 1-4. This part of the book treats with some thoroughness the notions of fundamental group and covering space, along with their many and varied applications. Some of the chapters of Part II are independent of one another; the dependence among them is expressed in the following diagram:
Certain sections of the book are marked with an asterisk; these sections may be omitted or postponed with no loss of continuity. Certain theorems are marked similarly. Any dependence of later material on these asterisked sections or theorems is indicated at the time, and again when the results are needed. Some of the exercises also depend on earlier asterisked material, but in such cases the dependence is obvious.
Sets of supplementary exercises appear at the ends of several of the chapters. They provide an opportunity for exploration of topics that diverge somewhat from the main thrust of the book; an ambitious student might use one as a basis for an independent paper or research project. Most are fairly self-contained, but the one on topological groups has as a sequel a number of additional exercises on the topic that appear in later sections of the book.
Possible course outlines. Most instructors who use this text for a course in general topology will wish to cover Chapters 1-4, along with the Tychonoff theorem in Chapter 5. Many will cover additional topics as well. Possibilities include the following:
the Stone-Cech compactification (38), metrization theorems (Chapter 6), the Peanocurve (44), Ascoli's theorem (45 and/or 47), and dimension theory (50). I have,in different semesters, followed each of these options.
For a one-semester course in algebraic topology, one can expect to cover most ofPart II.
It is also possible to treat both aspects of topology in a single semester, although with some corresponding loss of depth. One feasible outline for such a course would consist of Chapters 1-3, followed by Chapter 9; the latter does not depend on the material of Chapter 4. (The non-asterisked sections of Chapters 10 and 13 also are independent of Chapter 4.)
