Introduction to General Topology by K. D. Joshi: A Book Review
Topology is one of the most fascinating and abstract branches of mathematics, dealing with the properties of shapes and spaces that are preserved under continuous deformations. In this book, K. D. Joshi provides a comprehensive and rigorous introduction to the subject, covering topics such as sets and functions, metric spaces, topological spaces, continuity, compactness, connectedness, separation axioms, product spaces, quotient spaces, uniform spaces, category theory and selected topics.
The book is intended for undergraduate and graduate students who have some background in set theory, logic and real analysis. The author assumes that the reader is familiar with the basic concepts and results of these fields, and does not spend much time on reviewing them. Instead, he focuses on developing the theory of topology in a clear and logical manner, with plenty of examples and exercises to illustrate and reinforce the concepts. The book also contains some historical notes and references to other sources for further reading.
One of the strengths of the book is its emphasis on the connections between topology and other areas of mathematics, such as analysis, algebra and geometry. The author shows how many results and techniques from topology can be applied to solve problems in these fields, and vice versa. For example, he discusses how the Urysohn lemma can be used to prove the Tietze extension theorem, how the Stone-Äech compactification can be used to study convergence of sequences of functions, how the Brouwer fixed point theorem can be used to prove the existence of solutions to differential equations, and how category theory can be used to study functors and natural transformations.
Another strength of the book is its level of difficulty and depth. The book is not a mere introduction to topology, but a thorough and advanced treatment of the subject. The author does not shy away from presenting some of the most challenging and elegant results in topology, such as the Tychonoff theorem, the Urysohn metrization theorem, the Nagata-Smirnov metrization theorem, the embedding theorem for uniform spaces, and the duality theorem for compact Hausdorff spaces. The exercises are also well-designed and challenging, ranging from simple applications to original proofs and extensions of theorems.
The book is not without its drawbacks, however. One of them is its availability and accessibility. The book was published in 1983 by New Age International, a publisher based in India, and it is not easy to find a copy of it outside India. Moreover, the book is quite expensive compared to other textbooks on topology. Another drawback is its style and presentation. The book is written in a formal and concise manner, which may appeal to some readers but may also deter others who prefer a more informal and verbose approach. The book also lacks diagrams and illustrations that could help visualize some of the concepts and proofs.
Overall, Introduction to General Topology by K. D. Joshi is a remarkable book that covers a lot of ground in topology with rigor and elegance. It is suitable for readers who are looking for a challenging and comprehensive introduction to the subject, and who are willing to invest time and effort in mastering it. It is not recommended for readers who are looking for a quick and easy overview of topology, or who are not comfortable with abstract reasoning and formal language.