This volume studies topological spaces and continuous structures in full generality.
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 Product topology 3.3 Box topology 3.4 Quotient topology 3.5 Examples
Part II. Separation and Countability
Chapter 4. Separation Axioms
4.1 T₀, T₁, T₂ (Hausdorff) 4.2 Regular and normal spaces 4.3 Urysohn lemma 4.4 Tietze extension theorem 4.5 Examples
Chapter 5. Countability Conditions
5.1 First and second countability 5.2 Lindelöf property 5.3 Separability 5.4 Relations between properties 5.5 Examples
Chapter 6. Compactness
6.1 Definitions 6.2 Equivalent formulations 6.3 Compactness in product spaces 6.4 Applications 6.5 Examples
Part III. Connectedness and Convergence
Chapter 7. Connected Spaces
7.1 Connectedness 7.2 Path-connectedness 7.3 Components 7.4 Applications 7.5 Examples
Chapter 8. Convergence
8.1 Sequences 8.2 Nets 8.3 Filters 8.4 Limit points 8.5 Applications
Chapter 9. Continuity Revisited
9.1 Uniform continuity 9.2 Continuous extensions 9.3 Compact-open topology 9.4 Applications 9.5 Examples
Part IV. Advanced Constructions
Chapter 10. Function Spaces
10.1 Topologies on function spaces 10.2 Compact-open topology 10.3 Convergence structures 10.4 Applications 10.5 Examples
Chapter 11. Quotients and Identification Spaces
11.1 Equivalence relations 11.2 Quotient maps 11.3 Identification spaces 11.4 Applications 11.5 Examples
Chapter 12. Product and Limit Constructions
12.1 Infinite products 12.2 Inverse limits 12.3 Direct limits 12.4 Applications 12.5 Examples
Part V. Special Classes of Spaces
Chapter 13. Metric Spaces
13.1 Metrics and induced topologies 13.2 Completeness 13.3 Compactness 13.4 Examples 13.5 Applications
Chapter 14. Compact and Locally Compact Spaces
14.1 Local compactness 14.2 One-point compactification 14.3 Applications 14.4 Examples 14.5 Connections
Chapter 15. Paracompactness and Metrization
15.1 Definitions 15.2 Metrization theorems 15.3 Partition of unity 15.4 Applications 15.5 Examples
Part VI. Topological Structures in Analysis
Chapter 16. Topological Groups
16.1 Definitions 16.2 Continuity of operations 16.3 Examples 16.4 Applications 16.5 Connections
Chapter 17. Uniform Spaces
17.1 Uniform structures 17.2 Uniform continuity 17.3 Completeness 17.4 Applications 17.5 Examples
Chapter 18. Topological Vector Spaces
18.1 Definitions 18.2 Locally convex spaces 18.3 Duality 18.4 Applications 18.5 Examples
Part VII. Applications
Chapter 19. Analysis Applications
19.1 Continuity and compactness 19.2 Functional analysis links 19.3 Convergence structures 19.4 Applications 19.5 Examples
Chapter 20. Geometry Applications
20.1 Manifolds as topological spaces 20.2 Embedding problems 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Computational and Applied Topology
21.1 Topological data analysis (overview) 21.2 Shape reconstruction 21.3 Applications 21.4 Examples 21.5 Tools
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Set-theoretic topology 22.2 Descriptive topology 22.3 Infinite-dimensional topology 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Metrization challenges 23.2 Compactness questions 23.3 Set-theoretic issues 23.4 Computational aspects 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of topology 24.2 Key contributors 24.3 Evolution of concepts 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Topological property summary B. Common examples C. Proof techniques checklist D. Construction templates E. Cross-reference to other MSC branches