Topics

- basic topological notions
- examples of topological spaces
- separation properties
- classification of 2−manifolds
- introduction to homotopy theory

Outline/Homework

- Mon August 25: introduction, definition of topology
- Wed August 27: basis for a topology, interior, closure, limit points
- Fri August 29: metric spaces, metric topologies
**HW 1**: due Wednesday, September 3 (L^{a}T_{e}X template) SOLUTION- Mon September 1:
**No Class**(Labor Day) - Wed September 3: continuous mappings
- Fri September 5: subspaces
**HW 2**: due Wednesday, September 10 (L^{a}T_{e}X template) SOLUTION- Mon September 8: product spaces
- Wed September 10: connectedness
- Fri September 12: connected components and local connectedness
**HW 3**: due Wednesday, September 17 (L^{a}T_{e}X template) SOLUTION- Mon September 15: path connectedness and path components
- Wed September 17: compactness
- Fri September 19: compactness continued, finite intersection property
**HW 4**: due Wednesday, September 24 (L^{a}T_{e}X template) SOLUTION- Mon September 22: Tychonoff theorem
- Wed September 24: function spaces
- Fri September 26: separation axioms
**HW 5**: due Wednesday, October 1 (L^{a}T_{e}X template) SOLUTION- Mon September 29: Hausdorff, regular and normal spaces
- Wed October 1: Urysohn's lemma
- Fri October 3: Tietze extension theorem
**HW 6**: due Wednesday, October 8 (L^{a}T_{e}X template) SOLUTION- Mon October 6: Urysohn's metrization theorem
- Wed October 8: Complete metric spaces, Baire spaces
- Fri October 10: Peano's space filling curve
**HW 7**: due Wednesday, October 15 (L^{a}T_{e}X template) SOLUTION- Mon October 13: inverse systems
- Wed October 15: characterization of the Cantor set
- Fri October 17: characterization of the Cantor set continued
**HW 8**: due Wednesday, October 29 (L^{a}T_{e}X template) SOLUTION- Mon October 20:
**No Class**(Fall Break) - Wed October 22: quotient spaces
- Fri October 24: manifolds
- Mon October 27: embeddings of manifolds
- Wed October 29: simplicial and PL complexes
- Fri October 31: mapping class groups (Matt Durham)
**HW 9**: due Wednesday, November 5 (L^{a}T_{e}X template) SOLUTION- Mon November 3: polygons in
**R**² - Wed November 5: Schönflies theorem for polygons in
**R**² - Fri November 7: Jordan curve theorem
**HW 10**: due Monday, November 17 (L^{a}T_{e}X template) SOLUTION- Mon November 10: Jordan curve theorem continued
- Wed November 12: Euler characteristic
- Fri November 14: ??
- Mon November 17: classification of compact, connected 2−manifolds
- Wed November 19: classification of compact, connected 2−manifolds continued
- Fri November 21: classification of compact, connected 2−manifolds continued
**HW 11**: due Monday, December 1 (L^{a}T_{e}X template) SOLUTION- Mon November 24: introduction to homotopy
- Wed November 26:
**No Class**(Thanksgiving Holiday) - Fri November 28:
**No Class**(Thanksgiving Holiday) - Mon December 1: criteria for homotopy equivalence
- Wed December 3: homotopy extension property
- Fri December 5: the fundamental group
**HW 12**: due Wednesday, December 10 (L^{a}T_{e}X template)- Mon December 8: applications of the fundamental group
- Wed December 10: the induced homomorphism
- Mon December 15:
**FINAL EXAM**