Math 121: Introduction to Topology
|
| Date | Topic | Homework
|
| 1/8 (Mon) | Section 1.1: Definition of metric, open and closed sets |
A. Verify that R^n with the usual metric is a metric
space (the verification of the triangle inequality is 1.1
#3 below). 1.1: 2,3,5,7 (Optional: 1,4,8) |
| 1/10 (Wed) | Sections 1.2 and 1.3: Completeness, review of real line |
1.2: 3 (Optional: 7) 1.3: 4,7 |
| 1/12 (Fri) | Section 1.4: Products, R^n |
1.4: 1,3 |
| 1/15 |
University Holiday (MLK
Jr Day) |
|
| 1/17 |
Section 1.6: Continuous functions | 1.6: 1,2,3,4 (Optional: 5) |
| 1/19 |
Section 1.5: Compactness, Day I | A. Prove that a closed subset of a compact metric space
is compact. B. Prove that a finite union of compact subsets of a metric space is compact. 1.5: 1,2,4 (in part (b), assume that all the U_\alpha's are proper subsets of X),5 (Optional: 8) |
| 1/22 | Section 1.5: Compactness, Day II | 1.6: 7,8,9 (Optional: 10) |
| 1/24 |
Sections 2.1 and 2.2: Topological spaces,
subspaces |
2.1: 2,3,12 (Optional: 10) |
| 1/26 |
Section 2.3: Continuous functions |
A. Show that the induced topology is the smallest
topology which makes f:X \to Y continuous. 2.2: 1 2.3: 3,6,12 |
| 1/29 | Section 2.4: Basis for a topology | 2.4: 1,4,6 |
| 1/31 |
Section 2.10: Finite product spaces Section 2.13: Quotient spaces |
2.10: 5(a),(b) (Optional: (c),(d)) A. Prove directly (without using Theorem 13.4) that the spaces [0,1]/~ and [0,1]^2/~ described in class with the quotient topologies are homeomorphic to S^1 and S^1\times S^1, respectively. |
| 2/2 |
Section 2.5: Separation axioms, Day I | 2.13: 6 (prove this without using Theorem 13.4) 2.5: 1,2,3 |
| 2/5 | Section 2.5: Separation axioms, Day II | Due date for HW assigned
this week is Wed 2/14 2.5: 4,9 (Optional: 8) |
| 2/7 |
Section 2.6: Compactness | 2.6: 3,4,6 (Optional: 8) |
| 2/9 |
Section 2.7: One-point compactification | 2.7: 2,3,6,7 |
| 2/12 | Midterm Exam | midterm
info sample midterm problems |
| 2/14 |
Section 2.8: Connectedness | Due date for HW assigned this
week is Wed 2/21 2.8: 1,2,3,4,5,6 |
| 2/16 |
Section 2.9: Path-connectedness | 2.9: 1,2,3,4,6,7 |
| 2/19 | University Holiday (Presidents' Day) | |
| 2/21 |
Section 2.11: Zorn's lemma | 2:11: 3 |
| 2/23 |
Section 2.12: Infinite product spaces | 2.12: 3,4,7,8,9,11 |
| 2/26 |
Section 3.1: Groups Section 3.2: Homotopy of paths |
3.1: 1,4 |
| 2/28 | Section 3.2: More on homotopy of paths | 3.2: 1,2,4 |
| 3/1 |
Section 3.3: Fundamental group |
3.2: 5 3.3: 1,2,3,4 |
| 3/4 |
Section 3.4: Induced homomorphism |
3.3: 5,6,7 |
| 3/6 |
Section 3.7: Homotopy of maps Section 3.5: Covering spaces, Day I |
3.4: 1,2,3 2.13: 8 |
| 3/8 |
Section 3.5: Covering spaces, Day II |
3.5: 1,3,4,5 |
| 3/11 |
Section 3.5: Calculation of \pi_1(S^1) |
Don't need to turn in, but responsible for this
material on final exam. 3.5: 8,9,19 3.7: 5,7 |
| 3/13 |
Section 3.6: Applications |
|
| 3/15 |
Sections 3.8 and 3.9: Applications Review/summary |
|
| 3/18 (Mon) 8-11am |
Final
Exam (in our usual classroom) |
final
exam info sample final exam problems |