A taste of topology

Description

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. Title......Page 2 Preface......Page 5 Contents......Page 7 List of Symbols......Page 9 Introduction......Page 11 1.1 Sets and Functions......Page 14 1.2 Cardinals......Page 22 1.3 Cartesian Products......Page 26 Remarks......Page 29 2.1 Definitions and Examples......Page 32 2.2 Open and Closed Sets......Page 37 2.3 Convergence and Continuity......Page 43 2.4 Completeness......Page 49 2.5 Compactness for Metric Spaces......Page 61 Remarks......Page 68 3.1 Topological Spaces—Definitions and Examples......Page 70 3.2 Continuity and Convergence of Nets......Page 81 3.3 Compactness......Page 88 3.4 Connectedness......Page 98 3.5 Separation Properties......Page 109 Remarks......Page 116 4.1 Urysohn’s Lemma and Applications......Page 118 4.2 The Stone–Cech Compactification......Page 125 4.3 The Stone–Weierstraß Theorems......Page 130 Remarks......Page 138 5.1 Homotopy and the Fundamental Group......Page 141 5.2 Covering Spaces......Page 156 Remarks......Page 162 A The Classical Mittag-Leffler Theorem Derived from Bourbaki’s......Page 165 B Failure of the Heine–Borel Theorem in Infinite-Dimensional Spaces......Page 169 C The Arzela–Ascoli Theorem......Page 172 References......Page 175 Index......Page 177