Refined table of contents

Sections included in the syllabus (pensum) are emphasized in boldface.

Chapter 1: Set Theory and Logic

1. Fundamental Concepts
2. Functions
3. Relations
4. The Integers and the Real Numbers
5. Cartesian Products
6. Finite Sets
7. Countable and Uncountable Sets
8. * The Principle of Recursive Definition
9. Infinite Sets and the Axiom of Choice
10. Well-Ordered Sets
11. * The Maximum Principle

Chapter 2: Topological Spaces and Continuous Functions

12. Topological Spaces
13. Basis for a Topology (omitting lower limit and K-topologies)
14. The Order Topology
15. The Product Topology on X x Y
16. The Subspace Topology
17. Closed Sets and Limit Points
18. Continuous Functions
19. The Product Topology (omitting box topology)
20. The Metric Topology
21. The Metric Topology (continued)
22. * The Quotient Topology

Chapter 3: Connectedness and Compactness

23. Connected Spaces
24. Connected Subspaces of the Real Line
25. * Components and Local Connectedness
26. Compact Spaces
27. Compact Subspaces of the Real Line
28. Limit Point Compactness
29. Local Compactness

Chapter 4: Countability and Separation Axioms

30. The Countability Axioms
31. The Separation Axioms
32. Normal Spaces
33. The Urysohn Lemma
34. The Urysohn Metrization Theorem
35. * The Tietze Extension Theorem (omitting proof)
36. * Imbeddings of Manifolds

Chapter 5: The Tychonoff Theorem

37. The Tychonoff Theorem (omitting proof)
38. The Stone-Cech Compactification

Chapter 6: Metrization Theorems and Paracompactness

39. Local Finiteness
40. The Nagata-Smirnov Metrization Theorem
41. Paracompactness
42. The Smirnov Metrization Theorem

Chapter 7: Complete Metric Spaces and Function Spaces

43. Complete Metric Spaces
44. * A Space-Filling Curve
45. Compactness in Metric Spaces
46. Pointwise and Compact Convergence
47. Ascoli's Theorem

Chapter 8: Baire Spaces and Dimension Theory

48. Baire Spaces
49. * A Nowhere-Differentiable Function
50. Introduction to Dimension Theory

Chapter 9: The Fundamental Group

51. Homotopy of Paths
52. The Fundamental Group
53. Covering Spaces
54. The Fundamental Group of the Circle
55. Retractions and Fixed Points
56. * The Fundamental Theorem of Algebra
57. * The Borsuk-Ulam Theorem
58. Deformation Retracts and Homotopy Type
59. The Fundamental Group of S^n
60. Fundamental Groups of Some Surfaces

Chapter 10: Separation Theorems in the Plane

61. The Jordan Separation Theorem
62. * Invariance of Domain
63. The Jordan Curve Theorem
64. Imbedding Graphs in the Plane
65. The Winding Number of a Simple Closed Curve
66. The Cauchy Integral Formula

Chapter 11: The Seifert-van Kampen Theorem

Chapter 13: Classification of Covering Spaces

Chapter 12: Classification of Surfaces