Pure and Applied Mathematics: A Wiley-Interscience Series of #83: Topology: Point-Set and Geometric

The essentials of point-set topology, complete with motivation and numerous examples

Topology: Point-Set and Geometric presents an introduction to topology that begins with the axiomatic definition of a topology on a set, rather than starting with metric spaces or the topology of subsets of Rn. This approach includes many more examples, allowing students to develop more sophisticated intuition and enabling them to learn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors.

Along with the standard point-set topology topics—connected and path-connected spaces, compact spaces, separation axioms, and metric spaces—Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students.

Topology: Point-Set and Geometric features:

• A short introduction in each chapter designed to motivate the ideas and place them into an appropriate context
• Sections with exercise sets ranging in difficulty from easy to fairly challenging
• Exercises that are very creative in their approaches and work well in a classroom setting
• A supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs

Shick (mathematics and computer science, John Carroll U., Ohio) presents an excellent textbook for a first course in topology such as the junior or senior course required for all mathematics majors at his home institution. It is based on his belief that an introductory course has to cover enough point-set topology to prepare students for analysis geometry and other areas, but should also include more intuitive aspects, a big theorem as central reference, and the axiomatic definition of a topology on a set.
Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)

Shick (mathematics and computer science, John Carroll U., Ohio) presents an excellent textbook for a first course in topology such as the junior or senior course required for all mathematics majors at his home institution. It is based on his belief that an introductory course has to cover enough point-set topology to prepare students for analysis geometry and other areas, but should also include more intuitive aspects, a big theorem as central reference, and the axiomatic definition of a topology on a set. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)

This text covers the essentials of point-set topology in a relatively terse presentation, with lots of examples and motivation along the way. Along with the standard point-set topology topics (connected spaces, compact spaces, separation axioms, and metric spaces), the author includes path-connectedness, and a chapter on constructing spaces from other spaces (including products, quotients, etc.). The text culminates in to two main chapters, each independent of the other: 1) The Classification Theorem for Compact, Connected Surfaces and2) Fundamental Groups and Covering Spaces, with Applicationsgiving the reader the choice of which subject best suits them.

PAUL L. SHICK, PhD, is Professor in the Department of Mathematics and Computer Science at John Carroll University in Cleveland, Ohio. He earned a PhD in Mathematics from Northwestern University in 1984, working in the area of stable homotopy theory. He remains active in research in algebraic topology.

Foreword.

Acknowledgments.

1. Introduction: Intuitive Topology.

2. Background on Sets and Functions.

3. Topological Spaces.

4. More on Open and Closed Sets and Continuous Functions.

5. New Spaces from Old.

6. Connected Spaces.

7. Compact Spaces.

8. Separation Axioms.

9. Metric Spaces.

10. The Classification of Surfaces.

11. Fundamental Groups and Covering Spaces.

References.

Index.