Lecture Notes on General Topology

✍ Scribed by Guoliang Wang

• Publisher: World Scientific Publishing
• Year: 2021
• Tongue: English
• Leaves: 153
• Category: Library

✦ Synopsis

This book is intended as a one-semester course in general topology, a.k.a. point-set topology, for undergraduate students as well as first-year graduate students. Such a course is considered a prerequisite for further studying analysis, geometry, manifolds, and certainly, for a career of mathematical research. Researchers may find it helpful especially from the comprehensive indices. General topology resembles a language in modern mathematics. Because of this, the book is with a concentration on basic concepts in general topology, and the presentation is of a brief style, both concise and precise. Though it is hard to determine exactly which concepts therein are basic and which are not, the author makes efforts in the selection according to personal experience on the occurrence frequency of notions in advanced mathematics, and to related books that have received admirable reviews. This book also contains exercises for each chapter with selected solutions. Interrelationships among concepts are taken into account frequently. Twelve particular topological spaces are repeatedly exploited, which serve as examples to learn new concepts based on old ones.

✦ Table of Contents

Contents
Chapter 1 Introduction
Exercises 1
Selected Solutions 1
Chapter 2 Topological Spaces
2.1. Topological structures
2.2. Subspace topology
2.3. Covers
2.4. Point position with respect to a set
2.5. Metrics and the metric topology
Exercises 2
Selected Solutions 2
Chapter 3 Continuous Maps and Homeomorphisms
3.1. Continuous maps
3.2. Homeomorphisms
3.3. Topological properties
Exercises 3
Selected Solutions 3
Chapter 4 Connectedness
4.1. Connected spaces
4.2. Path-Connectedness
Exercises 4
Selected Solutions 4
Chapter 5 Separation and Countability Axioms
5.1. Axioms T0, T1, T2, T3, and T4
5.2. Hausdorff spaces
5.3. Regular spaces and normal spaces
5.4. Countability axioms
Exercises 5
Selected Solutions 5
Chapter 6 Compactness
6.1. Compact spaces
6.2. Interaction of compactness with other topological properties
6.3. Gromov-Hausdorff distance
Exercises 6
Chapter 7 Product Spaces and Quotient Spaces
7.1. Product spaces
7.2. Quotient spaces
Exercises 7
Selected Solutions 7
Appendix A Some Elementary Inequalities
Bibliography
Author Index
Subject Index