Lecture notes on topology and geometry - Huynh Quang Vu

Lecture notes on Topology and Geometry Huynh Quang Vu Faculty of Mathematics and Informatics, University of Natural Sciences, Vietnam National University, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam E-mail address: hqvu@hcmuns.edu.vn URL: http://www.math.hcmuns.edu.vn/~hqvu Abstract. This is a set of lecture notes prepared for a series of introductory courses in Topology and Geometry for undergraduate students at the University of Natural Sciences, Vietnam National University in Ho Chi Minh City. This is indeed a lecture notes in the sense that it is written to be delivered by a lecturer, namely myself, tailored to the need of my own students. I did not write it with self-study readers or with other lecturers in mind. Most statements are intended to be exercises. I provide proofs for some of the more dicult propositions, but even then there are still many details for students to fill in. More discussions will be carried out in class. I hope in this way I will be able to keep this lecture note shorter and more readable. In general if you encounter an unfamiliar notion, look at the Index and the Contents. A problem with a sign * is considered more dicult. A problem with a sign yis an important one. This lecture notes will be continuously developed and I intend to keep it freely available on my web page. Although at this moment it is only a draft I do hope that it is useful to the readers. Your comments are very welcomed. May 14, 2005 – September 23, 2008. Contents Part 1. General Topology 1 Chapter 1. Theory of Infinite Sets 3 1.1. The Cardinality of a Set 3 1.2. The Axiom of Choice 8 Guide for Further Reading in General Topology 10 Chapter 2. Topological Spaces 11 2.1. Topological Spaces 11 2.2. Continuity 15 2.3. Subspaces 18 Chapter 3. Connectivity 21 3.1. Connected Space 21 3.2. Path-connected Spaces 25 Further Reading 28 Chapter 4. Separation Axioms 29 4.1. Separation Axioms 29 Chapter 5. Nets 31 5.1. Nets 31 Chapter 6. Compact Spaces 35 6.1. Compact Spaces 35 Chapter 7. Product Spaces 39 7.1. Product Spaces 39 7.2. Tikhonov Theorem 42 Further Reading 44 Chapter 8. Compactifications 45 8.1. Alexandro Compactification 45 8.2. Stone-Cech Compactification 47 Chapter 9. Urysohn Lemma and Tiestze Theorem 49 9.1. Urysohn Lemma and Tiestze Theorem 49 Further Reading 53 Chapter 10. Quotient Spaces 55 10.1. Quotient Spaces 55 Chapter 11. Topological Manifolds 61 11.1. Topological Manifolds 61 Further Reading 63 iii iv CONTENTS Part 2. Algebraic Topology 65 Chapter 12. Homotopy and the fundamental groups 67 12.1. Homotopy and the fundamental groups 67 Chapter 13. Classification of Surfaces 69 13.1. Introduction to Surfaces 69 13.2. Classification Theorem 70 13.3. Proof of the Classification Theorem 71 Part 3. Dierential Topology 75 Chapter 14. Dierentiable manifolds 77 14.1. Smooth manifolds 77 14.2. Tangent spaces – Derivatives 80 Chapter 15. Regular Values 83 15.1. Regular Values 83 15.2. Manifolds with Boundary 87 Chapter 16. The Brouwer Fixed Point Theorem 91 16.1. The Brouwer Fixed Point Theorem 91 Chapter 17. Oriented Manifolds – The Brouwer degree 93 17.1. Orientation 93 17.2. Brouwer Degree 94 17.3. Vector Fields 98 Further Reading 99 Part 4. Dierential Geometry 101 Guide for Reading 102 Chapter 18. Regular Surfaces 103 18.1. Regular Surfaces 103 18.2. The First Fundamental Form 105 18.3. Orientable Surfaces 106 18.4. The Gauss Map 107 Bibliography 109 Index 111 Part 1 General Topology ... - tailieumienphi.vn