A First Course in Topology
豆瓣
Continuity and Dimension (Student Mathematical Library)
John McCleary
简介
How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincaré argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century.
The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension.
This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study.
目录
Cover 1
Title 4
Copyright 5
Contents 6
Introduction 8
Chapter 1. A Little Set Theory 14
Equivalence relations 16
The Schröder-Bernstein Theorem 20
The problem of Invariance of Dimension 22
Chapter 2. Metric and Topological Spaces 28
Continuity 37
Chapter 3. Geometric Notions 42
Chapter 4. Building New Spaces from Old 58
Subspaces 59
Products 65
Quotients 69
Chapter 5. Connectedness 80
Path-connectedness 90
Chapter 6. Compactness 96
Chapter 7. Homotopy and the Fundamental Group 108
Chapter 8. Computations and Covering Spaces 124
Chapter 9. The Jordan Curve Theorem 142
Gratings and arcs 143
The index of a point not on a Jordan curve 154
A proof of the Jordan Curve Theorem 159
Chapter 10. Simplicial Complexes 164
Simplicial mappings and barycentric subdivision 172
Chapter 11. Homology 184
Homology and simplicial mappings 192
Topological invariance 194
Where from here? 211
Bibliography 214
Notation Index 220
Subject Index 222
A 222
B 222
C 222
D 222
E 222
F 222
G 223
H 223
I 223
J 223
K 223
L 223
M 223
N 223
O 223
P 223
Q 223
R 223
S 223
T 224
U 224
V 224
Back Cover 226