Basic Topology 1

Basic Topology 1 by Avishek Adhikari and Mahima Ranjan Adhikari is the first in a series of three books intended as an introduction to various aspects of topology and its applications to other fields of mathematics. The present volume is dedicated to metric spaces and general topology, with Volume 2 covering topological groups, manifolds, and Lie groups, while Volume 3 will cover algebraic topology and the topology of fiber bundles. With an encyclopedic range of topics and terse exposition, Basic Topology 1 may make a reasonable reference for self-motivated learners, but will likely leave many students searching for more guidance.

 

The topics begin with prerequisites on sets, abstract algebra, and real analysis in Chapter 1. Upon first reading this section, I was excited to see the authors mention category theory as a topic introduced here as well. However, basic terms like “category” and “functor” are not actually defined anywhere in the book! Results are still occasionally restated in the language of category theory, but category theory does not play a significant role in the development overall (as in Topology: A Categorical Approach by Bradley, Bryson, and Terilla, for example). After the Prerequisites, Chapter 2 covers metric spaces and normed linear spaces before moving on to general topological spaces and continuous maps in Chapter 3. Then, Chapter 4 covers the separation axioms, with compactness and connectedness treated in Chapter 5. From there, Chapter 6 focuses on real-valued continuous functions, while Chapter 7 discusses countability, separability, and embeddings. Finally, Chapter 8 is dedicated to some historical details on topology as a field of study. Each of Chapters 2--8 culminate with about 40--50 exercises, including a handful (5--8) of multiple choice problems. Overall, Chapters 1--7 cover all of the topics that you would expect in a typical first course on point-set topology (and more), while Chapter 8 is an interesting addition that attempts to offer a brief historical account of various concepts and theorems presented in the text.

 

In terms of expositional style, the authors describe the book in the Preface as follows:

 

 

The discussion is, indeed, very straightforward. After a short summary of the main ideas, most sections proceed swiftly through the relevant definitions, examples, and theorems. Examples and proofs tend to be terse, presenting the necessary components of the argument while leaving a detailed verification to the reader. Ultimately, this encourages an active role on the part of the reader and, along with the light exposition, means that the text can cover a wide array of topics. However, I imagine that some care would need to be taken in the selection and presentation of topics if one were to use this textbook in an introductory course on topology, even at the beginning graduate level.

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Timothy Clark is an Associate Professor at Adrian College. His area of research is algebraic topology.