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Sheldon W. Davis
Publication Date: 
Number of Pages: 
The Walter Rudin Student Series in Advanced Mathematics
[Reviewed by
Michele Intermont
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There's another introductory undergraduate topology text on the block —  Topology, by Sheldon Davis. The book is divided into two halves. The first half is a basic course in point set topology, ending with a short chapter on homotopy. This would be suitable for a one-semester undergraduate course. The second part turns the book into a year-long course suitable for graduate students, according to the author. Personally, I would expect more from a course suitable for graduate students.

The second half treats the same topics as the first, at a slightly more advanced level. As one example, the product topology for countably many spaces is defined in Chapter 5 and in Chapter 15 for products over arbitrary indexing sets. In Chapter 5, there are three theorems which follow the definition including that the projection maps are open, continuous functions, and that a map into a product space is continuous iff its composition with each projection map is continuous. In Chapter 15 these two theorems are proven, and in addition, the weak topology is introduced and used.

In truth, this book is perfectly suitable for a first course in general topology, but I found little to enthuse me. It stands out from the crowd only in its inclusion of some more advanced set theory than most books provide, and a few more examples.

Michele Intermont teaches at Kalamazoo College.


Part I

1 Sets, Functions, Notation

2 Metric Spaces

3 Continuity

4 Topological Spaces

5 Basic Constructions: New Spaces From Old

6 Separation Axioms

7 Compact Spaces

8 Locally Compact Spaces

9 Connected Spaces

10 Other Types of Connectivity

11 Continua

12 Homotopy

Part II

13 A Little More Set Theory

14 Topological Spaces II

15 Quotients and Products

16 Convergence

17 Separation Axioms II

18 Compactness and Countability Properties

19 Stone-Cech Compactification

20 Paracompact Spaces

21 Metrization