学微积，用拓扑，越用拓扑越明白，不做糊涂人
2019-03-31 03:22阅读：
学微积，用拓扑，越用拓扑越明白，不做糊涂人
大家知道，现代微积分建立在欧几里德拓扑之上。因此，学微积，用拓扑，是当然之理。
因此，学微积，用拓扑，而且，越用拓扑越明白，不做糊涂人。
对于拓扑学必须把有“敬畏之心”。拓扑学不是小儿科，请见本文附件（全文在“无穷小微积分”网站）。
袁萌 陈启清
3月300日
附件：
Introduction to Topology Winter
2007（必须使用200冬季发布的这个电子版！）
Contents
1 Topology 9
1.1 Metric Spaces . 9
1.2 Open Sets (in a metric space) . . . . . . 10
1.3 Closed Sets (in a metric space) . . . . . 11
1.4 Topological Spaces
. . . . . . 11
1.5 Closed Sets (Revisited) . .
12
1.6 Continuity 13
1.7 Introduction to Topology.
14
1.8 Homeomorphism Examples . .. 16
1.9 Theorems On Homeomorphism . . 18
1.10 Homeomorphisms Between Letters of Alphabet . . ..
19
1.10.1 Topological Invariants . . . 19
1.10.2 Vertices . . . . 19
1.10.3 Holes . . . . . .. 20
1.11 Classication of Letters . . .
21
1.11.1 The curious case of the “Q”
22
1.12 Topological Invariants . . .. . 23
1.12.1 Hausdor Property . . . 23
1.12.2 Compactness Property
24
1.12.3 Connectedness and Path Connectedness Properties . . .
25
2 Making New Spaces From Old 27
2.1 Cartesian Products of Space
27
2.2 The Product Topology .
28
2.3 Properties of Product Spaces . . 29
3
2.4 Identication Spaces . . . .. 30
2.5 Group Actions and Quotient Spaces
34
3 First Topological Invariants 37
3.1 Introduction . 37
3.2 Compactness . . . 37
3.2.1 Preliminary Ideas . . . . . .. . . 37
3.2.2 The Notion of Compactness . . .. 40
3.3 Some Theorems on Compactnes
. 43
3.4 Hausdor Spaces . . . .47 3.5 T1 Spaces . .. ..
49
3.6 Compactication . .. . 50
3.6.1 Motivation . . . 50
3.6.2 One-Point Compactication . .
50
3.6.3 Theorems . . 51
3.6.4 Example 55
3.7 Connectedness . . 57
3.7.1 Introduction . 57
3.7.2 Connectedness . . . . .
58
3.7.3 Path-Connectedness . . 61
4 Surfaces 63
4.1 Surfaces . . . . . . . . . 63
4.2 The Projective Plane . . . . .
63
4.2.1 RP2 as lines in R3 or a sphere with antipodal points
identied. . . . . . . 63 4.2.2 The Projective Plane as a Quotient
Space of the Sphere . . . . 65
4.2.3 The Projective Plane as an identication space of a disc
. . . . . . 66
4.2.4 Non-Orientability of the Projective Plane . . . . .. .
69 4.3 Polygons 69
4.3.1 Bigons . . 71
4.3.2 Rectangles . . . . 72
4.3.3 Working with and simplifying polygons . . .
74
4.4 Orientability . 76
4.4.1 Denition . . 76
4
4.4.2 Applications To Common Surfaces . .
77
4.4.3 Conclusion . . . . . . 80
4.5 Euler Characteristicn. .. .80
4.5.1 Requirements . . 80
4.5.2 Computatio. . . 81
4.5.3 Usefulness . . . . 83
4.5.4 Use in identication polygons . . . . . .
83
4.6 Connected Sums . . 85
4.6.1 Denition . . . 85
4.6.2 Well-denedness . . 85
4.6.3 Examples . . .. . 87
4.6.4 RP2#T= RP2#RP2#RP2 . .88
4.6.5 Associativity . . 90
4.6.6 Eect on Euler Characteristic . . . . . .
90
4.7 Classication Theorem . . .
92
4.7.1 Equivalent denitions . . . . .
92
4.7.2 Proof . . . . . 93
5 Homotopy and the Fundamental Group 97 5.1 Homotopy of
functions . . . . 97
5.2 The Fundamental Group . .
100
5.2.1 Free Groups . . . 100
5.2.2 Graphic Representation of Free Group . .. .
101
5.2.3 Presentation Of A Group . . . . . 103
5.2.4 The Fundamental Group .. . 103
5.3 Homotopy Equivalence between Spaces . . . . . 105 5.3.1
Homeomorphism vs. Homotopy Equivalence . 105
5.3.2 Equivalence Relation . . . . .
106
5.3.3 On the usefulness of Homotopy Equivalence
106
5.3.4 Simple-Connectedness and Contractible spaces . . . .
107
5.4 Retractions . . . . 108
5.4.1 Examples of Retractions . . . . . 108
5
5.5 Computing the Fundamental Groups of Surfaces: The
Seifert-Van Kampen Theorem . . .
110
5.5.1 Examples: . . . 112
5.6 Covering Spaces . . . 113
5.6.1 Lifting . . 117
