Customer Reviews

Introduction to Metric and Topological Spaces

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I purchased Introduction to Metric and Topological Spaces two years ago. I was unprepared for its rigor. I am not a mathematics major, but I enjoy reading mathematics. My background includes calculus, linear algebra, differential equations, and other applied mathematics, but I have not had a course in real analysis. W. A. Sutherland intended this text as the next step after analysis.

After a brief foray, I retreated, placed Sutherland back on my bookshelf, and attacked some marginally easier introductory texts: Metric Spaces by Victor Bryant, Introduction to Topology by Bert Mendelson, and most recently, several chapters in Introduction to Analysis by Maxwell Rosenlicht. I periodically return to W. A. Sutherland's text to measure my understanding. I am now working on chapter five, Compact Spaces.

I doubt that Introduction to Metric and Topological Spaces would be foreboding to students that are familiar with real analysis. Sutherland understands that the abstractness and generalization can be difficult and shows concern with motivating the student. He repeatedly attempts to illustrate the value of generalization, especially in the study of continuity.

Sutherland often uses a lengthy series of examples of increasing difficulty to illustrate abstract concepts. In his discussion of metric spaces, we begin with Euclidian n-space metrics, and move on to discrete metric spaces, function spaces, and even Hilbert sequence spaces. He introduces open sets and topological spaces in a similar fashion.

The author occasionally suggests that the student might wish to make a geometrical diagram to help clarify some subtle point, but Sutherland includes few geometrical drawings in his text. His focus is clearly on proofs using the axioms of metric spaces and topological spaces.

Sutherland highlights sections that either require more knowledge of abstract algebra, or for other reasons are thought to be more severe.

Despite Sutherland's use of Introduction in the title, I suggest that any reader considering independent study might defer tackling Introduction to Metric and Topological Spaces until after completing a more basic text. Possibly a better title might be A Second Introduction to Metric and Topological Spaces.

A lot of books on topology assume some basic knowledge of real analysis, which can throw a lot of readers off. This book starts from the very beginning, and thus is truly a great introduction. Each section has some good exercises, with even a few pointers at the back of the book for the more challenging ones. It starts with topological aspects, and then refers to them in the case of metric spaces (amongst many others), which is a much better approach than most other books, as the reader doesn't take the details of the specific to the general. A great little book, which is a must for most advanced maths Analysis courses.

I enjoyed reading this book because of its clarity, conciseness, and nice way of relating topological and metric spaces. This book is ideal for the student who is learning about these subjects for the first time, whether or not they intend to do more advanced work on the subject. The reader who wants to go on and learn about more advanced topics, should consult Munkres's book.