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