Topology

Topology

For a senior undergraduate or first year graduate-level course in Introduction to Topology. Appropriate for a one-semester course on both general and algebraic topology or separate courses treating each topic separately.

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

[11+] Supplementary Exercises: Well-Ordering

13. Basis for a Topology

13. Classification of Covering Spaces

16. The Subspace Topology

17. Closed Sets and Limit Points

18. Continuous Functions

19. The Product Topology

20. The Metric Topology

21. The Metric Topology (continued)

22. The Quotient Topology

[22+] Supplementary Exercises: Topological Groups

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

29. Local Compactness

30 The Countability Axioms

31 The Separation Axioms

32 Normal Spaces

33 The Urysohn Lemma

34 The Urysohn Metrization theorem

36 Imbeddings of Manifolds

51 Homotopy of Paths

52 The Fundamental Group

53 Covering Spaces

54 The Fundamental Group of the Circle

58 Deformation Retracts and Homotopy Type

59 The Fundamental Group of $\mathbf{S}^n$

60 Fundamental Groups of Some Surfaces

67 Direct Sums of Abelian Groups

68 Free Products of Groups

71 The Fundamental Group of a Wedge of Circles

73 The Fundamental Groups of the Torus and the Dunce Cap

74 Fundamental Groups of Surfaces

75. Homology of Surfaces