#topology
57. Manifolds and Cell Complexes
This volume studies manifolds and combinatorial models such as CW complexes. It connects topology, geometry, and algebraic topology through structured spaces. Part I. Topological Manifolds Chapter 1. Definitions and Examples 1.1 Topological manifolds 1.2 Local Euclidean structure 1.3 Charts and atlases 1.4 Examples 1.5 Basic properties Chapter 2. Maps Between Manifolds 2.1 Continuous maps 2.2 Homeomorphisms 2.3 Embeddings 2.4 Submanifolds 2.5 Examples Chapter 3. Topological Properties 3.1 Compactness 3.2 Connectedness...
55. Algebraic Topology
This volume studies topological spaces through algebraic invariants. It translates geometric and continuous structure into groups, rings, modules, and homological data. Part I. Fundamental Constructions Chapter 1. Spaces and Maps 1.1 Topological spaces 1.2 Continuous maps 1.3 Homotopy 1.4 Homotopy equivalence 1.5 Examples Chapter 2. Fundamental Group 2.1 Paths and loops 2.2 Homotopy classes 2.3 Definition of π₁ 2.4 Functoriality 2.5 Examples Chapter 3. Covering Spaces 3.1 Covering maps 3.2...
54. General Topology
This volume studies topological spaces and continuous structures in full generality. It provides the foundational language for modern analysis and geometry. Part I. Topological Spaces Chapter 1. Basic Definitions 1.1 Topologies and open sets 1.2 Closed sets 1.3 Bases and subbases 1.4 Examples 1.5 Constructions Chapter 2. Continuous Functions 2.1 Definitions 2.2 Equivalent formulations 2.3 Homeomorphisms 2.4 Embeddings 2.5 Examples Chapter 3. Subspaces and Product Spaces 3.1 Subspace topology 3.2...
22. Topological Groups, Lie Groups
This volume studies groups equipped with topology and smooth structure. It connects algebra, topology, and geometry, with applications in analysis and physics. Part I. Topological Groups Chapter 1. Topological Groups 1.1 Definitions and examples 1.2 Continuity of multiplication and inversion 1.3 Subgroups and quotient groups 1.4 Homomorphisms 1.5 Basic properties Chapter 2. Fundamental Properties 2.1 Compactness 2.2 Connectedness 2.3 Local compactness 2.4 Separation axioms 2.5 Examples Chapter 3. Haar Measure...