Synopses & Reviews
Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level.
To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics.
After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces.
Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student.
Basic treatment covers preliminaries (sets, relations, etc.), topological spaces, continuous functions (mappings) and homeomorphisms, special types of topological spaces, metric spaces, more. Geometric and axiomatic approach for easier accessibility. Exercises. Bibliography.
Undergraduate-level treatment covers preliminaries, topological spaces, continuous functions (mappings) and homeomorphisms, metric spaces, more. Exercises. Bibliography.
This basic treatment, specially designed for undergraduates, covers preliminaries sets, relations, and more topological spaces, continuous functions mappings and homeomorphisms, special types of topological spaces, metric spaces, and more. The book utilizes a geometric and axiomatic approach for easier accessibility. Includes exercises and a bibliography.
Includes bibliographical references (p. 145-146) and index.
Table of Contents
CHAPTER 0 PRELIMINARIES
3. The Algebra of Sets
4. Euler-Venn Diagrams
6. Infinite Sets
7. Miscellaneous Assumptions Regarding the Real Numbers
CHAPTER 1 TOPOLOGICAL SPACES-BASIC DEFINITIONS AND THEOREMS
1. Neighborhood Systems and Topologies
2. Open Sets in a Topological Space
3. Limit Points and the Derived Set
4. The Closure of a Set
5. Closed Sets
7. Limits of Sequences; Hausdorff Spaces
8. Comparison of Topologies
9. "Bases, Countability Axions, Separability"
10. "Sub-bases, Product Spaces"
CHAPTER 2 CONTINUOUS FUNCTIONS (MAPPINGS) AND HOMEOMORPHISMS
2. Continuous Functions (Mappings)
4. Product Spaces
CHAPTER 3 VARIOUS SPECIAL TYPES OF TOPOLOGICAL SPACES (VARIETIES OF COMPACTNESS)
1. Compact Spaces
2. Separation Axioms
3. Countable Compactness
4. Local Compactness
CHAPTER 4 FURTHER SPECIAL TYPES OF TOPOLOGICAL SPACES (MOSTLY VARIETIES OF CONNECTEDNESS)
2. Connected Spaces
4. Local Connectedness
5. Arcwise Connectedness
CHAPTER 5 METRIC SPACES
2. Some Properties of Metric Spaces
3. Metrization Theorems
4. Complete Metric Spaces
5. Category Theorems