| Week | Topics | |
|---|---|---|
| 1 | Aug 23 Aug 25 Aug 27 |
§7 Countable and uncountable sets §12 Topological spaces Standard topology on R and Cantor set |
| 2 | Sep 3 Sep 5 |
§13 Basis for topology Subbasis, §14 Order topology |
| 3 | Sep 8 Sep 10 Sep 12 |
§15 Product topology on XxY, §16 Subspace topology §17 Closed sets and limit points, Hausdorff property
|
| 4 | Sep 15 Sep 17 Sep 19 |
§18 continuous functions homeomorphisms, embeddings §19 Product toplogy |
| 5 | Sep 22 Sep 24 Sep 26 |
§20 metric topology §21 uniform convergence |
| 6 | Sep 29 Oct 1 Oct 3 |
§22 Quotient topology §23 Connectedness §24 Connected subspaces of R |
| 7 | Oct 6 PRELIM Oct 8 Oct 10 |
§25 Components and path connectedness §26 Compact spaces |
| 8 | Oct 15 Oct 17 |
§27 Compact subspaces of the reals (video lecture) |
| 9 | Oct 20 Oct 22 Oct 24 |
§28 Sequential compactness Sequential compactness of countable products; Second countability and separability §51 Homotopy |
| 10 | Oct 27 Oct 29 Oct 31 |
§51 Path homotopy, groups §52 Fundamental group §53 Covering spaces |
| 11 | Nov 3 Nov 5 Nov 7 |
§53 Covering spaces §54 Fundamental group of the circle §55 Retractions and Brouwer fixed point theorem |
| 12 | Nov 10 Nov 12 Nov 14 |
§58 Deformation retracts, homotopy equivalence §59 Fundamental groups of spheres |
| 13 | Nov 17 Nov 19 Nov 21 |
§60 Fundamental groups of products, eg torus Fundamental group of punctured torus (free product) and §60 projective plane Definition and examples of manifolds |
| 14 | Nov 24 | Kinsey Ch.4. Classification of surfaces (see Kinsey "Topology of Surfaces" for this material; available through Cornell for no charge) |
| 15 | Dec 1 NO CLASS Dec 3 Dec 5 |
Kinsey Ch.4. Finish proof of classification of surfaces Kinsey Ch. 5. Euler characteristic and proof of invariance (Note: Kinsey proof of Theorem 5.13 is incomplete. To complete it, one can separately distinguish the surfaces, eg by puncturing and using the homotopy equivalence to a wedge of circles.) |
| 16 | Dec 8 | Comments on proof of invariance of Euler characteristic |
| Dec 19, 9am | FINAL |