Elements of topology (Q2838296)

The book under review provides an introduction to the basics of general topology and (non-homological) algebraic topology. As the author points out in the preface, the text is based on his experiences in teaching topology courses to both undergraduate and graduate students at the University of Delhi, India, and elsewhere over several years. All the material presented here has been tested in courses for students who have an elementary knowledge of analysis, algebra and group theory, and has been found quite accessible to those audiences. Apart from these moderate prerequisites, enough background material from set theory and the algebra of number systems has been collected in two appendices to make the book largely self-contained.NEWLINENEWLINE As to the precise contents, the book consists of fifteen chapters and the just mentioned two appendices.NEWLINENEWLINE Chapter 1 introduces topological structures and topological spaces, together with their conceptual framework and their most general properties. Chapter 2 discusses continuous maps between topological spaces, including homeomorphisms and the notion of topological equivalence, and turns then to products of topological spaces and more general induced topologies. Chapter 3 is devoted to the concept of connectedness in general topology, with extra sections on connected spaces, connected components of an arbitrary topological space, path-connected spaces and local connectivity, respectively. The notion of convergence of sequences in topological spaces is dealt with in Chapter 4, where nets, filters, ultrafilters, and the introduction of Hausdorff spaces are the main topics to develop the appropriate framework in this context. Chapter 5 explains the classical countability axioms for topological spaces, with special emphasis on first and second countable spaces, separable space, and Lindelöf spaces.NEWLINENEWLINE Compactness properties of topological spaces are subsequently treated in Chapter 6, including the study of compact spaces, countably compact spaces, compact metric spaces, locally compact spaces, compactifications, and proper topological maps. Chapter 7 presents various topological constructions such as quotient spaces and identification maps, cones, suspensions and joins, wedge sums and smash products, attaching maps and adjunction spaces, coinduced and coherent topologies with respect to families of topological spaces, and compactly generated spaces. These first seven chapters constitute the first part of the book and -- covering nearly one half of the material -- can be considered as the core of the text. This part is suitable for a one-semester first course in general topology for advanced undegraduates, as the author explicitly points out in the preface. The following Chapters 8 through 11 form the second part of the book, which is devoted to some more specific and advanced topics in point-set topology, partly with a special view toward applications in (functional) analysis. More precisely, Chapter 8 gives an introduction to the various separation axioms for topological spaces, with a special focus on regular spaces, normal spaces, completely regular spaces, and the Stone-Čech compactification of the latter. Chapter 9 treats paracompact spaces as generalized compact spaces, on the one hand, and the Nagata-Smirnov metrization theorem on the other. Complete metric spaces, completions of metric spaces, and Baire spaces are briefly touched upon in Chapter 10, while the subsequent Chapter 11 is devoted to the study of function spaces and their topologies, especially with regard to the topology of pointwise convergence, the topology of uniform convergence, the compact-open topology, and the topology of compact convergence in certain spaces of continuous mappings.NEWLINENEWLINE The next part of the book is comparatively short. Consisting of the following two Chapters 12 and 13, this part describes the basic properties of topological groups and some elementary facts about topological transformation groups. Group actions on topological spaces, their orbit spaces, and illustrating examples by means of classical linear groups and their quotient spaces conclude the discussion presented in this third part of the book. Finally, the remaining Chapters 14 and 15 make up the fourth (and last) part of this textbook. They are meant to provide the first steps into the realm of algebraic topology, mainly by explaining the notions of fundamental groups and covering spaces of topological spaces. The study of these objects is carried out in some greater detail and includes the following topics. Chapter 14 deals with homotopic maps in general, contractible spaces, (deformation) retracts, the fundamental group of a topological space and its functorial properties, fundamental groups of spheres, the path and homotopy lifting properties, and the Seifert-van Kampen theorem. The necessary concepts and techniques from advanced group theory are provided in an extra section of this chapter, and several instructive applications of elementary homotopy theory are presented as well. Chapter 15 discusses topological covering maps, local homeomorphisms, lifting properties of covering maps, the monodromy theorem, universal covering spaces, deck transformations of covering maps, proper actions of discrete groups and their associated covering spaces, and the existence theorem for universal covering spaces of certain topological spaces. Among the many concrete applications, the famous Borsuk-Ulam theorem is derived, too.NEWLINENEWLINE As already mentioned, there are two appendices at the end of the book. Appendix A provides (with proofs) the necessary background material from set theory, including orderings, ordinal numbers and cardinal numbers, whereas appendix B describes the basic properties of the fields of real numbers, complex numbers, and quaternions. Each section ends with a carefully composed list of related exercises, and the entire text is interspersed with numerous instructive, directly related examples and counterexamples as well as with many illuminating figures and diagrams. Together with the utmost lucid, detailed and didactically well-balanced presentation of the material, these special features make the book a suitable source for self-study, on the one hand, and for a profound course in topology on the other. Both students and instructors can profit a great deal from this excellent primer, which shows the author's rich teaching experience just as much as his expository skills throughout the book.