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Essential Topology

Martin D. Crossley
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Stephen T. Ahearn
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In Essential Topology, Martin Crossley strikes a good balance between the abstraction inherent in the standard topological definitions and the underlying geometric intuition of topology.

The first two parts of the text give a crisp, uncluttered, and well-flowing presentation of standard point-set topology topics. Covered in these two parts are continuity, topological spaces, subspaces, connectivity, compactness, the Hausdorff property, homeomorphisms, product spaces, and quotient spaces. Omitted are some favorite topics — metric spaces, space-filling curves, and the Jordan curve theorem, for example — but these omissions streamline the presentation.

The third part of the text introduces the basics of algebraic topology: homotopy, the Euler characteristic, homotopy groups, simplicial homology, singular homology, and fibre bundles, and could serve as a text for a second course in topology. (Does anyone teach such a course at the undergraduate level?) Unfortunately, the exposition on algebraic topology is written at a much more general and abstract level, making it more difficult to select a topic from algebraic topology to include at the end of a first course. The fundamental group, for example, is introduced as a special case of general homotopy groups. This increase in abstraction is a consequence, I believe, of the author's stated intent of writing a book to prepare undergraduate students for graduate study in algebraic topology — not a goal which I think serves most students well.

The exercises mostly concern concrete examples (show that the function f : S1RP1  is continuous...) with a few abstract exercises sprinkled throughout the text. There are very few exercises which ask the reader to prove general theorems such as "the image of a path-connected space is path connected." In fact, the point-set topology portion of the text proves too much. I would rather the author had left more of the standard facts as exercises for the students to do. (This comment reflects my bias toward teaching point-set topology using a modified Moore method.) Of course, an instructor could easily supplement the exercises.

The book is very readable and would be accessible to an undergraduate studying independently. Solutions to selected exercises are provided, something I would not like as an instructor of a course, but which might be helpful to a student reading the text on her own. The first two parts of the text are certainly accessible to undergraduates with sufficient mathematical maturity (a junior or senior). The third part requires knowledge of group theory, and, as I mentioned, is written at a more abstract level.

On the whole, Essential Topology is a nice addition to the introductory topology textbook literature. The text gives a clean presentation of basic point-set topology without neglecting the geometric intuition and without including too many topics. The only comparable text might be Czes Kosniowski's A First Course in Algebraic Topology, which is unfortunately out of print and uses basic group theory throughout the text. I will probably use Essential Topology the next time I teach topology (if I decide against using the Moore method).

Stephen T. Ahearn ( teaches mathematics at Macalester College in St. Paul, MN. His primary research interests are in algebraic topology and computational topology/geometry but allows himself to be distracted by other interesting topics as in his article "Tolstoy's Integration Metaphor from War and Peace.'' He also enjoys hiking, swimming, baking bread, and reading.

Introduction.- Continuous Functions.- Topological Spaces.- Interlude.- Topological Properties.- Deconstructionist Topology.- Interlude.- Homotopy.- The Euler Number.- Homotopy Groups.- Simplicial Homology.- Singular Homology.- More Deconstructionism.- Solutions to Selected Exercises.- Bibliography.- Index.