Contents v

§6.4. More Examples 69

§6.5. Simply Connected Spaces 70

§6.6. The Isomorphism Theorem 71

§6.7. The Bolzano–Weierstrass Theorem 72

§6.8. The Lifting Property 73

§6.9. Proof of the Isomorphism Theorem 74

Chapter 7. Existence of Universal Covers 79

§7.1. The Main Result 80

§7.2. The Covering Property 82

§7.3. Simple Connectivity 84

Part 2. Surfaces and Geometry

Chapter 8. Euclidean Geometry 87

§8.1. Euclidean Space 87

§8.2. The Pythagorean Theorem 90

§8.3. The X Theorem 91

§8.4. Pick’s Theorem 92

§8.5. The Polygon Dissection Theorem 96

§8.6. Line Integrals 98

§8.7. Green’s Theorem for Polygons 100

Chapter 9. Spherical Geometry 103

§9.1. Metrics, Tangent Planes, and Isometries 103

§9.2. Geodesics 105

§9.3. Geodesic Triangles 107

§9.4. Convexity 110

§9.5. Stereographic Projection 111

§9.6. The Hairy Ball Theorem 113

Chapter 10. Hyperbolic Geometry 115

§10.1. Linear Fractional Transformations 115

§10.2. Circle Preserving Property 116