Topology and Maps (International Scholars Forum)

I. Preliminaries.- 1. Fundamental Notions of Set Theory.- 2. Relations and Mappings.- 3. Partial and Linear Orderings; Cartesian Products.- 4. Lattices.- 5. Algebraic Structures.- 6. Categories and Functors.- II. Topological Spaces.- 7. Open and Closed Sets.- 8. Topologies and Neighborhoods.- 9. Limit Points.- 10. Bases and Subbases.- 11. First and Second Countable Spaces.- 12. Metric Spaces.- 13. Nets.- 14. Filters.- 15. Topologies Defined by Other Topologies.- Examples and Exercises.- III. Continuity and Separation Axioms.- 16. Continuous and Open Mappings.- 17. Topologies Defined by Mappings.- 18. Separation Axioms.- 19. Continuous Functions on Normal Spaces.- Examples and Exercises.- IV. Methods for Constructing New Topological Spaces from Old.- 20. Subspaces.- 21. Topological Sums.- 22. Topological Products.- 23. Quotient Topology and Quotient Spaces.- 24. Projective and Inductive Limits.- Examples and Exercises.- V. Uniform Spaces.- 25. Uniformities and Topologies.- 26. Uniformity and Separation Axioms.- 27. Uniformizable Spaces.- 28. Uniform Continuity and Uniform Spaces.- 29. Completeness in Uniform Spaces.- 30. Completeness, Compactness, and Completions.- 31. Topological Groups and Topological Vector Spaces.- 32. Metrizability.- 33. Fixed Points.- 34. Proximity Spaces.- Examples and Exercises.- VI. Compact Spaces and Various Other Types of Spaces.- 35. Compact Spaces.- 36. Countable Compactness and Sequential Compactness.- 37. Compactness in Metric Spaces.- 38. Locally Compact Spaces.- 39. MB-Spaces.- 40. k-Spaces and kr-Spaces.- 41. Baire Spaces.- 42. Pseudocompact Spaces.- 43. Paracompact Spaces.- 44. Compactifications.- Examples and Exercises.- VII. Generalizations of Continuous Maps.- 45. Almost Continuous Maps.- 46. Closed Graphs.- 47. Almost Continuity and Closed Graphs.- 48. Graphically Continuous Maps.- 49. Nearly Continuous and w-Continuous Maps.- 50. Semicontinuous Maps.- 51. Approximately Continuous Functions.- 52. Applications of Almost Continuity.- Examples and Exercises.- VIE. Function Spaces.- 53. The Set of All Maps.- 54. Compact-Open Topology and the Topology of Joint Continuity.- 55. Subsets of FE with Induced Topologies.- 56. The Uniformities on FE.- 57. 𝔖-Uniformities and 𝔖-Topologies.- 58. Equicontinuous Maps.- 59. Equicontinuity and Metric Spaces.- 60. Sequential Convergence in Function Spaces.- Examples and Exercises.- IX. Extensions of Mappings.- 61. Extensions of Maps on Completely Regular and Metric Spaces.- 62. The Hahn-Banach Extension Theorem.- 63. A General Extension Theorem.- Examples and Exercises.- X. C(X) Spaces.- 64. Stone-Weierstrass Theorem.- 65. Embeddings of X into C(X).- 66. C(X) Spaces for Compact Spaces X.- 67. Separability in C(X).- 68. C(X) Spaces for Completely Regular Spaces X.- 69. Characterization of Banach and Fréchet Spaces C(X).- 70. Characterization of Locally Convex Spaces C(X).- Epilogue.- Examples and Exercises.