Summary and Info
This skinny little math book from the Springer Universitext series achieves excellence on many levels. First of all, anyone familiar with the old quip "A topologist is someone who cannot tell the difference between a coffee mug and a donut" will instantly smile when they see the cover. The exposition is downright beautiful, and the organization of the material could not be more perfect. The remarkable thing is that the examples not only demonstrate the concepts, but also play a large role in the development. The choice of fonts and notation is well thought-out and, although minor, contributes greatly to the excellence of the book. One of the best features of this book is its length. With less than 200 pages, one can reasonably set a goal to read it cover to cover. The well-chosen examples not only aid in understanding, but also serve to introduce the reader to concepts from other areas of mathematics. On that note, not only those seeking an introduction to topology, but also anyone new to advanced mathematics, and in addition seasoned mathematicians who are thinking about writing books themselves, will benefit greatly from reading this book.The author divides the material into five chapters-- 1. Set Theory, 2. Metric Spaces, 3. Topological Spaces, 4. Function Spaces, and 5. Basic Algebraic Topology. There are a number of good exmples from chapters 2 and 3 that serve to compare and contrast properties of metric spaces and topological spaces, as can be expected in any topology text, however the examples used here are interesting in their own right in other areas of math. The author uses the Zariski topology on the prime ideals of a commutative ring in many places. The reader will also meet various function spaces and see how pointwise vs. uniform convergence manifest themselves through suitably chosen topologies.A number of unique features worth noting are the proof of the Baire category theorem, which is derived from the so called Mittag-Leffler theorem (this is probably the only introductory text which proves this), and Tychonoff's theorem is proved using nets by expressing compactness as every net has a convergent subnet. Also of interest are proofs of the Stone-Weierstrass theorem and the Arzela-Ascoli theorem. On top of all this, there is still some room left at the end to introduce some basic homotopy theory. The fundamental group is defined and covering spaces are introduced. The author proves that homotopy-equivalent spaces have isomorphic fundamental groups, shows that paths and path homotopies can be lifted, and uses this to establish that the fundamental group of the circle is isomorphic to the integers. This is used to prove the Brouwer fixed-point theorem.
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