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.
This work is suitable for undergraduate students as well as advanced students and research workers. It consists of ten chapters, the first six of which are meant for beginners and are therefore suitable for undergraduate students; Chapters VII-X are suitable for advanced students and research workers interested in functional analysis. This book has two special features: First, it contains generalizations of continuous maps on topological spaces, e. g. , almost continuous maps, nearly continuous maps, maps with closed graph, graphically continuous maps, w-continuous maps, and a-continuous maps, etc. and some of their properties. The treatment of these notions appears here, in Chapter VII, for the first time in book form. The second feature consists in some not-so-easily-available nuptial delights that grew out of the marriage of topology and functional analysis; they are topics mainly courted by functional analysts and seldom given in topology books. Specifically, one knows that the set C(X) of all real- or com plex-valued continuous functions on a completely regular space X forms a locally convex topological algebra, a fortiori a topological vector space, in the compact-open topology. A number of theorems are known: For example, C(X) is a Banach space iff X is compact, or C(X) is complete iff X is a kr-space, and so on. Chapters VIII and X include this material, which, to the regret of many interested readers has not previously been available in book form (a recent publication (Weir [ 6]) does, however, contain some material of our Chapter X).
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