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Introduction to Metric and Topological Spaces

Edition: 
2
Publisher: 
Oxford University Press
Number of Pages: 
206
Price: 
39.95
ISBN: 
9780199563081

This is a brief, clearly-written introduction to point-set topology. The approach is axiomatic and abstract — the development is motivated by a desire to generalize properties of the real numbers rather than a need to solve problems from other areas of mathematics. In particular there is very little mention of function spaces, although some of the examples deal with the existence of solutions to integral and differential equations as an application of Banach’s fixed-point theorem.

The book assumes some familiarity with the topological properties of the real line, in particular convergence and completeness. The level of abstraction moves up and down through the book, where we start with some real-number property and think of how to generalize it to metric spaces and sometimes further to general topological spaces. Most of the book deals with metric spaces.

The book has modest goals. It introduces the most important concepts of topology but does not take any of them very far. The exercises at the end of each chapter are partly routine applications of the chapter contents and partly extensions into more difficult areas not covered in the chapter. There is a companion web site that has solutions to all the exercises, as well as a great deal of supplemental material that did not fit into the main narrative. Because the book starts out with the real line, it is slanted somewhat towards analysis. Its aim is topology and it is not as nearly as thorough as analysis-oriented books such as Wilansky’s Topology for Analysis or Kelley’s General Topology. It also has an interesting chapter on quotient spaces, focused on Moebius strips and tori with various numbers of holes.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning.

 


 

Date Received: 
Thursday, September 24, 2009
Reviewable: 
Include In BLL Rating: 
Wilson A. Sutherland
Publication Date: 
2009
Format: 
Paperback
Audience: 
Category: 
Textbook
Allen Stenger
05/19/2010
BLL Rating: 
1. Introduction

2. Notation and terminology

3. More on sets and functions
Direct and inverse images
Inverse functions

4. Review of some real analysis
Real numbers
Real sequences
Limits of functions
Continuity
Examples of continuous functions

5. Metric spaces
Motivation and definition
Examples of metric spaces
Results about continuous functions on metric spaces
Bounded sets in metric spaces
Open balls in metric spaces
Open sets in metric spaces

6. More concepts in metric spaces
Closed sets 
Closure 
Limit points 
Interior 
Boundary 
Convergence in metric spaces 
Equivalent metrics 
Review 

7. Topological spaces
Definition 
Examples 

8. Continuity in topological spaces; bases
Definition 
Homeomorphisms 
Bases 

9. Some concepts in topological spaces

10. Subspaces and product spaces
Subspaces 
Products 
Graphs 
Postscript on products

11. The Hausdorff condition
Motivation 
Separation conditions

12. Connected spaces
Motivation
Connectedness
Path-connectedness
Comparison of definitions
Connectedness and homeomorphisms

13. Compact spaces
Motivation
Definition of compactness
Compactness of closed bounded intervals
Properties of compact spaces
Continuous maps on compact spaces
Compactness of subspaces and products
Compact subsets of Euclidean spaces
Compactness and uniform continuity
An inverse function theorem

14. Sequential compactness
Sequential compactness for real numbers
Sequential compactness for metric spaces

15. Quotient spaces and surfaces
Motivation 
A formal approach 
The quotient topology 
Main property of quotients 
The circle
The torus
The real projective plane and the Klein bottle
Cutting and pasting
The shape of things to come

16. Uniform convergence
Motivation 
Definition and examples 
Cauchy's criterion 
Uniform limits of sequences 
Generalizations 

17. Complete metric spaces
Definition and examples
Banach's fixed point theorem
Contraction mappings
Applications of Banach's fixed point theorem

Bibliography

Index
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Wednesday, May 19, 2010

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