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General Toplogy: An Introduction

Tom Richmond
De Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Textbook
[Reviewed by
Timothy Clark
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This textbook covers the standard topics from an introductory course in point-set topology, along with more modern topics in topology related to applications in computer science. It is intended for students at the advanced undergraduate or beginning graduate level. Overall, the exposition is lucid and well motivated, incorporating a variety of useful examples and an excellent assortment of illustrations, with a refreshing choice of topics that also manages to hit all the usual marks for a typical one-semester course in general topology.
Throughout the text, abstract concepts are motivated by concrete ideas and there are plenty of figures to aid visualization. There are over 740 exercises in all, with each section generally receiving somewhere in the range of 10-20 exercises. The author’s development is unique in its emphasis on order and equivalence. Though he is also sure to elucidate the more traditional view of topology as a study of continuity, nearness, and convergence, ordered sets and equivalence relations underpin much of the organization and exposition of topics. 
The book consists of 14 chapters, including the preliminary Chapter 0 that reviews essential concepts from set theory and elementary logic. Chapters 1-7 then give an account of the usual suspects from classical point-set topology.
The aforementioned emphasis on order becomes explicit in Chapters 8-11. Referencing modern considerations in computer science -- where we must utilize non-Hausdorff topologies on finite sets or otherwise pixelated representations of continuous phenomena -- Chapter 8 transitions to studying Alexandroff spaces (which include finite topological spaces) and their connection to quasi-orders (reflexive, transitive relations). Chapter 9 studies lattices, especially the lattice of topologies on a given set. Chapter 10 then introduces partially ordered topological spaces, while Chapter 11 explores generalizations of metrics to non-Hausdorff spaces, including connections to the Alexandroff topologies and quasi-orders studied earlier. 
Chapters 12 and 13 return, for the most part, to more traditional applications of topology to analysis and to establishing a foundation for algebraic topology, respectively. The focus in Chapter 12 is on uniform structures and their asymmetric extensions. Finally, Chapter 13 is dedicated to continuous deformations, developing the notion of homotopy between curves and investigating the fundamental group of a space.
A one-semester undergraduate course would likely aim to cover most of Chapters 1-7 (with a few omissions). Additional enrichment could be selected from the later chapters based on the instructor’s preference, if time permits. While this is not a textbook in applied mathematics, Chapters 8-13 could be used as a base for a more advanced course that features aspects of computational topology or other modern applications.
Timothy Clark is an Assistant Professor at Adrian College.  His area of research is algebraic topology.