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Table of Contents

1. Introduction to topology.- 1.1. An overview.- 2. Point-set topology in ?n.- 2.1. Open and closed sets in ?n.- 2.2. Relative neighborhoods.- 2.3. Continuity.- 2.4. Compact sets.- 2.5. Connected sets.- 2.6. Applications.- 3. Point-set topology.- 3.1. Open sets and neighborhoods.- 3.2. Continuity, connectedness, and compactness.- 3.3. Separation axioms.- 3.4. Product spaces.- 3.5. Quotient spaces.- 4. Surfaces.- 4.1. Examples of complexes.- 4.2. Cell complexes.- 4.3. Surfaces.- 4.4. Triangulations.- 4.5. Classification of surfaces.- 4.6. Surfaces with boundary.- 5. The euler characteristic.- 5.1. Topological invariants.- 5.2. Graphs and trees.- 5.3. The euler characteristic and the sphere.- 5.4. The euler characteristic and surfaces.- 5.5. Map-coloring problems.- 5.6. Graphs revisited.- 6. Homology.- 6.1. The algebra of chains.- 6.2. Simplicial complexes.- 6.3. Homology.- 6.4. More computations.- 6.5. Betti numbers and the euler characteristic.- 7. Cellular functions.- 7.1. Cellular functions.- 7.2. Homology and cellular functions.- 7.3. Examples.- 7.4. Covering spaces.- 8. Invariance of homology.- 8.1. Invariance of homology for surfaces.- 8.2. The Simplicial Approximation Theorem.- 9. Homotopy.- 9.1. Homotopy and homology.- 9.2. The fundamental group.- 10. Miscellany.- 10.1. Applications.- 10.2. The Jordan Curve Theorem.- 10.3. 3-manifolds.- 11. Topology and calculus.- 11.1. Vector fields and differential equations in ?n.- 11.2. Differentiable manifolds.- 11.3. Vector fields on manifolds.- 11.4. Integration on manifolds.- Appendix: Groups.- References.