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Undergraduate topology. A working textbook. (English)
Oxford: Oxford University Press (ISBN 978-0-19-870233-7/hbk; 978-0-19-870234-4/pbk). xii, 144~p. (2014).
The focus of this text is to help students build theory in point-set topology for themselves. In 140 pages, a student is guided through some of the major concepts in an undergraduate course: topological spaces, continuity and convergence, invariants, bases and products, separation axioms, in a way to encourage the Moore approach of teaching. The authors sought to develop mathematical arguments using chapters with short expositions on definitions, examples, propositions with warnings, followed by “expansions" that include some proofs or hints for proof. At the end of six chapters, in a section with 35 pages called “Essential Exercises", some of the topics are further explored and recommendations are made at the end of each chapter for students to try certain exercises. In addition to some usual examples of metric spaces that students have encountered in a real analysis course, certain examples of topological spaces are given, most at the beginning of the text, and exploited throughout the text to illustrate certain concepts: Cantor excluded-middle, cofinite topology on an infinite set, cocountable topology on an uncountable set, included-point (and excluded-point) topology based on a point on a set, the Arens-Fort space, and the intrinsic topology on a totally-ordered set. A major strength of the text is that generalizations are motivated. It would make a good text for instructors interested in students taking an active role in their learning, or for an “Independent Study" course when such a course is not offered. A mature student could use the book for a review.
Reviewer: Pao-Sheng Hsu (Columbia Falls)
Classification: I95 G95
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