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Topology


"Topology" is a branch of mathematics that studies the properties of "spaces" that stay the same under stretching, bending, or twisting, without tearing or gluing. It focuses on basic ideas like "continuity", "compactness", and "connectedness", helping mathematicians understand the structure and behavior of different shapes and surfaces. Topology is widely used in "mathematics", physics, and computer science to model and analyze objects in flexible ways. By exploring how points and sets relate to each other, topology provides a foundation for advanced topics in geometry, analysis, and dynamical systems, making it a key tool for both theoretical and applied studies.


You can explore a collection of "free topology books" that are available online for students and self-learners. These resources cover both "introductory and advanced topology" topics, providing accessible material to strengthen your understanding of the subject.

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Free Topology Books
Algebraic Topology - Allen Hatcher
"Algebraic Topology" explains how "algebraic topology" uses algebra to study shapes and spaces. Written by "Allen Hatcher", the book clearly covers "homotopy" and "homology", making complex ideas easier for advanced students to understand.
Solutions to Hatcher Algebraic Topology - M. Poulsen
This text provides clear "solutions" to exercises, focusing on "homology", "fundamental groups", and "cohomology". It helps students understand complex algebraic topology concepts step by step, making proofs, computations, and problem-solving accessible for both classroom learning and self-study.
Elementary Topology Problem Textbook - O.Ya.Viro, et al
This text is a "problem-based" guide to "topology", covering "topological spaces", "continuity", and the fundamental group. Through exercises and examples, it helps students understand abstract concepts, develop intuition, and build problem-solving skills, making it ideal for self-study or introductory courses in topology.
Foliations & Geometry of 3-Manifolds - Danny Calegari
This text studies "foliations" in "3-manifolds", showing how surfaces and laminations reveal their "topology" and geometry. The book connects dynamics, pseudo-Anosov flows, and modern manifold theory, making complex interactions in low-dimensional spaces accessible to researchers and advanced students.
General Topology - Pete L. Clark
This is a "concise" guide to the fundamentals of "topological spaces", "continuity", and "compactness". It introduces metric spaces, separation axioms, convergence, and constructions like product and subspace topologies. Clear and rigorous, it’s ideal for beginners seeking a solid foundation in modern topology.
Geometry with an Introduction to Cosmic Topology - PDF
This text is a student-friendly textbook exploring "non-Euclidean geometries", "hyperbolic geometry", and "cosmic topology". It links geometry with the universe’s shape, covering curvature, Möbius transformations, and parallax, offering exercises and essays for deep understanding, making it ideal for learning geometry in a cosmological context.
Introduction to Topology - Renzo Cavalieri
This text is a clear guide to "topology", covering "metric spaces", "topological spaces", and "continuity". It explains core concepts with examples and exercises, helping beginners understand how spaces behave, develop intuition for connectedness and compactness, and build a strong foundation for further study in mathematics.
Lectures on Differential Topology - Riccardo Benedetti
This text is a clear guide to "differential topology", focusing on "smooth manifolds", "smooth maps", and "transversality". With intuitive examples and exercises, it helps graduate students and researchers understand core concepts, apply geometric methods, and build a strong foundation in modern differential topology.
Lectures on Geometry of Manifolds - Liviu Nicolaescu
This text introduces "differential geometry" and "manifold theory" in a clear, accessible way. It explains tangent spaces, Riemannian metrics, curvature, and geodesics with examples and exercises, making it ideal for students and researchers learning "geometric analysis" and modern manifold concepts.
Manifold Theory - Peter Petersen
This text explores "smooth manifolds", "transversality", and "de Rham cohomology", explaining how differential forms and intersections reveal geometric and topological properties. The book makes advanced concepts in geometry and topology accessible to students and researchers, connecting theory with practical mathematical applications.
Manifolds - Current Research Areas - Paul Bracken
This text explores modern "manifold theory" and "differential geometry", presenting recent research on smooth manifolds, geometric structures, and topology. The book highlights current methods, problems, and applications, making advanced concepts accessible to researchers and students in contemporary mathematics.
Real Variables - Robert B. Ash
This is a clear and rigorous textbook introducing "real analysis" and "metric space" concepts. It covers limits, continuity, differentiation, integration, and topology, with examples and exercises for "self-study", helping students build a strong foundation for advanced mathematics.
Simplicial & Dendroidal Homotopy Theory - Gijs Heuts
This text introduces "homotopy theory" using "simplicial sets" and "dendroidal sets", bridging classical and modern approaches. It explains higher structures like 8-categories and 8-operads with combinatorial models, offering clear insight into spaces and algebraic operations up to homotopy.
Topological Groups: Yesterday, Today, Tomorrow - Morris
This textbook explores the history, current research, and future of "topological groups". It explains key results like "Lie groups", the Pontryagin-van Kampen "duality", and the Peter-Weyl Theorem, offering both beginners and experts a clear view of the theory’s development and ongoing research directions.
Topology: A Categorical Approach - Tai-Danae Bradley
This text is a modern guide to "topology", using "category theory" and "universal properties" to explain concepts like compactness and connectedness. With intuitive explanations and examples, it helps readers understand abstract structures and relationships in topology from a conceptual, categorical perspective.
Topology Lecture Notes - Ali Sait Demir
This text is a clear guide to "topology", covering "metric spaces", "topological spaces", and "continuity". With easy-to-follow definitions, examples, and exercises, it helps students build a strong foundation in point-set topology and understand how abstract concepts connect to real mathematical problems.
Topology of Numbers - Allen Hatcher
This text presents "number theory" through a "geometric" and "topological" viewpoint. It connects integers, modular arithmetic, and divisibility with spatial structures, helping readers visualize patterns and gain intuitive understanding of numbers, making complex arithmetic concepts clear and engaging.
Topology Without Tears - Sidney A. Morris
This text is a beginner-friendly guide to "topology", covering "topological spaces", "continuous functions", and "metric spaces". With clear explanations, examples, and exercises, it helps readers understand abstract concepts, develop intuition, and build a solid foundation for further study in point-set topology and mathematical analysis.
Topology for the Working Mathematician - Michael Muger
This text is a clear guide to "topology", focusing on "topological spaces", "continuous functions", and "metric spaces". With intuitive examples and explanations, it helps readers understand abstract concepts, explore key constructions like product and quotient topologies, and build a solid foundation in modern mathematics.

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