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无泪的拓扑 Topology Without Tears, 拓扑经典教材
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TOPOLOGY WITHOUT TEARS
1
SIDNEY A. MORRIS
2
Version of October 14, 2007
3
1
c
Copyright 1985-2007. No part of this book may be reproduced by any process without prior written permission
from the author. If you would like a printable version of this book please e-mail your name, address, and
commitment to respect the copyright (by not providing the password, hard copy or soft copy to anyone else) to
s.morris@ballarat.edu.au
2
A version of this book translated into Persian is expected to be available soon.
3
This book is being progressively updated and expanded; it is anticipated that there will be about fifteen chapters
in all. If you discover any errors or you have suggested improvements please e-mail: s.morris@ballarat.edu.au
Contents
0 Introduction 5
0.1 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
0.2 Readers – Locations and Professions . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.3 Readers’ Compliments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1 Topological Spaces 13
1.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Finite-Closed Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 The Euclidean Topology 35
2.1 Euclidean Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Basis for a Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Basis for a Given Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Limit Points 56
3.1 Limit Points and Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Homeomorphisms 70
4.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Non-Homeomorphic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Continuous Mappings 90
2
CONTENTS 3
5.1 Continuous Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 Metric Spaces 104
6.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
6.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.4 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Baire Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7 Compactness 149
7.1 Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 The Heine-Borel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8 Finite Products 162
8.1 The Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.2 Projections onto Factors of a Product . . . . . . . . . . . . . . . . . . . . . . . . 167
8.3 Tychonoff’s Theorem for Finite Products . . . . . . . . . . . . . . . . . . . . . . . 172
8.4 Products and Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.5 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9 Countable Products 182
9.1 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 The Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.3 The Cantor Space and the Hilbert Cube . . . . . . . . . . . . . . . . . . . . . . . 189
9.4 Urysohn’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.5 Peano’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10 Tychonoff’s Theorem 213
10.1 The Product Topology For All Products . . . . . . . . . . . . . . . . . . . . . . . 214
10.2 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.3 Tychonoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.4 Stone-
˘
Cech Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.5 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
4 CONTENTS
Appendix 1: Infinite Sets 247
Appendix 2: Topology Personalities 270
Appendix 3: Chaos Theory and Dynamical Systems 277
Appendix 4: Hausdorff Dimension 306
Appendix 5: Topological Groups 319
Bibliography 343
Index 359
Chapter 0
Introduction
Topology is an important and interesting area of mathematics, the study of which will not only
introduce you to new concepts and theorems but also put into context old ones like continuous
functions. However, to say just this is to understate the significance of topology. It is so
fundamental that its influence is evident in almost every other branch of mathematics. This
makes the study of topology relevant to all who aspire to be mathematicians whether their first
love is (or will be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics,
geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical
finance, mathematical modelling, mathematical physics, mathematics of communication, number
theory, numerical mathematics, operations research or statistics. (The substantial bibliography
at the end of this book suffices to indicate that topology does indeed have relevance to all these
areas, and more.) Topological notions like compactness, connectedness and denseness are as
basic to mathematicians of today as sets and functions were to those of last century.
Topology has several different branches — general topology (also known as point-set topology),
algebraic topology, differential topology and topological algebra — the first, general topology,
being the door to the study of the others. I aim in this book to provide a thorough grounding in
general topology. Anyone who conscientiously studies about the first ten chapters and solves at
least half of the exercises will certainly have such a grounding.
For the reader who has not previously studied an axiomatic branch of mathematics such as
abstract algebra, learning to write proofs will be a hurdle. To assist you to learn how to write
proofs, quite often in the early chapters, I include an
aside which does not form part of the proof
but outlines the thought process which led to the proof.
5
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