The Simplest Introductory Textbook on Topology

Recommended is Introduction to Topology, commonly translated into Chinese as 《拓扑学导论》. The author, Bert Mendelson, is a mathematics professor at Smith College. Originally written as an introductory lecture note for undergraduate topology students, it was later adapted into a book. It frequently tops the best-seller lists for topology books on Amazon.

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The main objective of this book is to provide a concise yet comprehensive overview of the fundamental topics concerning sets (or collections) with mathematical structure. The entire book is brief, totaling just over 200 pages, with clear exposition, a gradual progression in difficulty, and cleverly designed exercises—making it very accessible for beginners, especially self-learners. However, it requires readers to have a basic foundation in calculus and some familiarity with rigorous definitions and proofs. According to some readers with a solid mathematical background, it is possible to finish the entire book within a few days. On the other hand, some readers note that the content is neither comprehensive nor deeply detailed, but for a book of just over 200 pages, mastering point-set topology well is the main achievement.

The book succinctly yet comprehensively covers the fundamental topics of topology, starting from set theory and progressively delving into metric spaces and topological spaces, followed by connectedness and compactness. Initially, it was written as a textbook for a one-semester course aimed at undergraduates who have completed calculus and are familiar with theorems, definitions, and proofs.

Main topics covered in the book:

  1. Set Theory: Introduction to the fundamentals of sets.

  2. Metric Spaces: Concepts of metrics, distance functions, and common topologies.

  3. Topological Spaces: General topology including open and closed sets, neighborhoods, continuous mappings, and related concepts.

  4. Connectedness: How spaces are connected and criteria for connectedness.

  5. Compactness and Countability: Compact spaces, countable bases, countable covers, and related topics.

The book focuses primarily on point-set topology, laying the groundwork for further study in algebraic or other areas of topology. Readers can learn basic abstract structures as well as develop rigorous proof techniques.