Introduction to Topology

1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology 20 2.4. Gluing topologies 23 2.5. Proper maps 25 3. Connectedness 26 4. Separation axioms and the Hausdorff property 32 4.1. More on the Hausdorff property 34 5. Compactness and its relatives 35 5.1. Local compactness and paracompactness 41 5.2. Compactness in metric spaces 47 6. Quotient spaces 52 6.1. Quotient topology 52 6.2. Fibre products and amalgamated sums 59 6.3. Group actions on topological spaces 64 7. Homotopy 74 8. The fundamental group and some applications 79 8.1. Definition and basic properties 79 8.2. Applications 82 9. Covering spaces 87 10. Classification of covering spaces 97 References 102