Textbook in PDF format This book is an ambitious but optimistic treatment of the subject of topology. Not only does Elements of Topology: Theory and Practice, Second Edition treat the standard basic material on point-set topology, but it also gives an introduction to algebraic topology, a treatment of manifolds, a discussion of function spaces, and some ideas of knot theory. Even the exciting new topic of the Jones polynomial is covered. After discussing the key ideas of topology, the author examines the more advanced topics of algebraic topology and manifold theory. Topic coverage has been reduced in this second edition, and exercise sets have been added at the end of each section. The book as a whole - and the first two chapters in particular - offer many examples. Solutions to selected exercises are included at the end. Today’s students need a text that speaks to them in their own language, and at a pace with which they are comfortable. That is the goal of this edition. Taking a fresh and accessible approach to a venerable subject, this text provides excellent representations of topological ideas. It forms the foundation for further mathematical study in real analysis, abstract algebra, and beyond. Topology is by its very nature abstract. Traditionally it follows the Euclidean paradigm of formulating axioms and definitions and then proving theorems. The famous R. L. Moore method for teaching mathematics (which is still alive and well today) is based on the teaching of point-set topology, just because the essential structure of the subject lends itself well to a formulaic presentation. The purpose of the present book will be to present topology in a new way: as the natural evolution of ideas that the student has already seen in multivariable calculus, differential equations, and real analysis. Certainly we shall present the traditional concepts of topological space, open and closed set, the separation axioms, and so forth. All the basic cornerstones of the subject will xbe treated. But they will be presented in a familiar context, bolstered by examples and discussion. Our goal is to bring not only the mathematics student, but also the physics student and the engineeer and the computer scientist, up to speed in this evermore-important and rapidly growing subject area. After laying the groundwork for the fundamental notions of topology, we shall present applications of the ideas in Morse theory, manifold theory, and homotopy theory. An entree to homology theory will also be provided. The book will also have a generous selection of exercises at all levels. Certainly drill exercises are important so that the student can master the basic concepts. And thought and exploration problems are also necessary so that the student can stretch his/her abilities and look to the development of the ideas. Every chapter has drill exercises and “problems for exploration.” The former are self-explanatory. The latter are more difficult problems that will be used selectively