22 Best Topology Books

James Munkres introduces general and algebraic topology in two connected parts. The text develops topological spaces, continuity, connectedness, and compactness before exploring fundamental groups, covering spaces, surfaces, and applications, with examples and exercises throughout.

J. P. May presents algebraic topology for advanced graduate students, combining classical treatments with developments in the field. Chapters cover foundational topics, include problems for further exploration, and conclude with sketches of areas beyond the standard introductory course.

Allen Hatcher introduces algebraic topology through fundamental groups, covering spaces, homology, cohomology, and homotopy theory. Geometric explanations, examples, and exercises build intuition, while optional sections extend the main course into topics often omitted from introductory treatments.

M. A. Armstrong introduces point-set, geometric, and algebraic topology through the search for invariants of spaces. Illustrated explanations and exercises connect continuity, compactness, fundamental groups, surfaces, homology, knots, and covering spaces for students with prior mathematical preparation.

Stephen Willard introduces general topology through convergence, compactness, metrization, uniform spaces, and function spaces, alongside geometric topics involving connectedness and homotopy. Exercises introduce additional examples, while historical notes and references support further study of the subject.

Bert Mendelson develops the foundations of topology for undergraduates familiar with calculus proofs. Beginning with sets and metric spaces, the text progresses to topological spaces, connectedness, and compactness, with exercises supporting each stage.

John Milnor gives a concise introduction to differential topology, beginning with smooth manifolds and maps. He develops tangent spaces, orientations, vector fields, homotopy, and degree, with proofs of Sard's theorem and the Hopf theorem.

Victor Guillemin and Alan Pollack introduce differential topology using transversality as a central tool. Written for students with analysis and linear algebra, the text explores smooth manifolds, mappings, intersection theory, and related ideas through explanations and exercises.

Lynn Arthur Steen and J. Arthur Seebach organize topological examples to show which properties do and do not imply others. Explanations, diagrams, and exercises support the study of these relationships and the construction of further counterexamples.

John Kelley develops general topology as a foundation for modern analysis, offering both a graduate text and a reference. Integrated problems range from conceptual exercises and counterexamples to applications and small theories that extend the main discussion.

Klaus Jänich introduces topological spaces, continuity, quotient spaces, compactness, homotopy, and covering spaces through concise explanations and illustrations. The text connects foundational concepts with geometric constructions, while a supplementary chapter develops the set-theoretic background used in the discussion.

James Munkres develops algebraic topology through a concrete treatment of homology and cohomology. Chain complexes, simplicial methods, duality, and related theorems provide the framework, while examples and exercises connect algebraic constructions with the study of topological spaces.

Martin Crossley takes undergraduate readers from continuity and topological spaces to homotopy, homology, and fibre bundles. Examples motivate the main ideas, with discussions of the Euler number and the hairy ball theorem connecting definitions to recognizable mathematical problems.

George Simmons introduces topology and modern analysis through the related ideas of continuity and linearity. His textbook explores the mathematical structures behind these concepts, connecting geometric and analytical perspectives for readers developing their understanding of abstract mathematics.

Seymour Lipschutz presents a concise review of general topology supported by solved and supplementary problems. Step-by-step solutions and explanations help readers practice the subject's methods and reinforce material studied in a college course.

Fred Croom introduces topology with an emphasis on geometric intuition and applications to analysis. The text covers metric and topological spaces, continuity, connectedness, compactness, and related constructions, then introduces geometric, differential, and algebraic topics through exercises and examples.

Colin Adams introduces the mathematical study of knots through diagrams, examples, and exercises. Beginning with simple knots, he develops ways to distinguish and manipulate them, connecting accessible problems with invariants, surfaces, and questions from contemporary knot theory.

Charles Nash and Siddhartha Sen introduce topology and differential geometry for physics students. Manifolds, homotopy, homology, cohomology, fibre bundles, and Morse theory lead to applications involving defects, Yang-Mills theories, instantons, and monopoles.

Morris Hirsch introduces the topological ideas used to study differentiable manifolds, including immersions, embeddings, and Morse theory. Geometric explanations accompany exercises ranging from theorem applications to further theoretical developments, with attention to surfaces and cobordism.

C. R. F. Maunder develops elementary algebraic topology from a homotopy-theoretic viewpoint for advanced undergraduates and beginning graduate students. The text connects topological problems with algebraic methods, giving substantial attention to homotopy theory and duality in homology manifolds.

William Fulton introduces algebraic topology by emphasizing connections with other branches of mathematics. His first course develops ideas including winding numbers, fundamental groups, homology, and cohomology, using those relationships to motivate the subject's methods and applications.

Boris Dubrovin, Anatoly Fomenko, and Sergei Novikov develop the geometry and topology of manifolds in this second volume. The text expands modern geometric education beyond curves and surfaces, providing mathematical foundations useful in fields such as relativity.