About the Book
How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincaré argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century.
The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension.
This text is based on the author’s course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study.
Table of Contents
Introduction
Chapter 1. A Little Set Theory
Chapter 2. Metric and Topological Spaces
Chapter 3. Geometric Notions 29
Chapter 4. Building New Spaces from Old
Chapter 5. Connectedness
Chapter 6. Compactness
Chapter 7. Homotopy and the Fundamental Group
Chapter 8. Computations and Covering Spaces
Chapter 9. The Jordan Curve Theorem
Chapter 10. Simplicial Complexes
Chapter 11. Homology
Bibliography
Notation Index
Subject Index
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