Math 121: Introduction to Topology
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| Date | Topic | Homework
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| 1/3 (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: 1,2,3,4,5,7,8 |
| 1/5 (Wed) | Sections 1.2 and 1.3: Completeness, review of real line |
1.2: 5,7 1.3: 4,7 |
| 1/7 (Fri) | Section 1.4: Products, R^n |
1.4: 1,3,6 |
| 1/10 |
Section 1.6: Continuous
functions |
1.6: 1,2,3,4,5 |
| 1/12 |
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),5,8 |
| 1/14 |
Section 1.5: Compactness, Day II |
1.6: 7,8,9,10 |
| 1/17 | University Holiday (MLK Jr Day) | |
| 1/19 |
Sections 2.1 and 2.2: Topological spaces,
subspaces |
2.1: 2,3,10,12 |
| 1/21 |
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 |
| 1/24 | Section 2.4: Basis for a topology | 2.4: 1,4,6 |
| 1/26 |
Section 2.10: Finite product spaces Section 2.13: Quotient spaces |
2.10: 5(a)-(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. |
| 1/28 |
Section 2.5: Separation axioms, Day I | 2.13: 6 (prove this without using Theorem 13.4) 2.5: 1,2,3 |
| 1/31 | Section 2.5: Separation axioms, Day II | 2.5: 4,8,9 |
| 2/2 |
Section 2.6: Compactness | 2.6: 3,4,6,7,8 |
| 2/4 |
Section 2.7: One-point compactification | 2.7: 2,3,6,7 You can turn in
HW from this week on Wed 2/9. |
| 2/7 | Midterm Exam | Midterm
info Sample midterm problems |
| 2/9 |
Section 2.8: Connectedness | 2.8: 1,2,3,4,5,6 |
| 2/11 |
Section 2.9: Path-connectedness | 2.9: 1,2,3,4,6,7 |
| 2/14 | Section 2.11: Zorn's lemma | 2.11: 3 |
| 2/16 |
Section 2.12: Infinite product spaces | 2.12: 3,4,7,8,9,11 |
| 2/18 |
Section 3.1: Groups Section 3.2: Homotopy of paths |
3.1: 1,4 |
| 2/21 |
University Holiday (Presidents' Day) | |
| 2/23 | Section 3.2: More on homotopy of paths | 3.2: 1,2,4 |
| 2/25 |
Section 3.3: Fundamental group |
3.2: 5 3.3: 1,2,3,4 |
| 2/28 |
Section 3.4: Induced homomorphism |
3.3: 5,6,7 |
| 3/2 |
Section 3.7: Homotopy of maps Section 3.5: Covering spaces, Day I |
3.4: 1,2,3 2.13: 8 |
| 3/4 |
Section 3.5: Covering spaces, Day II |
3.5: 1,3,4,5 |
| 3/7 |
Section 3.5: Calculation of \pi_1(S^1) |
Don't need to turn in 3.5: 8,9,10 3.7: 5,7 |
| 3/9 |
Section 3.6: Applications |
|
| 3/11 |
Sections 3.8 and 3.9: Applications Review/summary |
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| 3/17 (Thu) | Final
Exam |
Final
info Sample final problems |