Math 121: Introduction to Topology

Date  Topic  Homework

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: Onepoint 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: Pathconnectedness  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 

3/17 (Thu)  Final
Exam 
Final
info Sample final problems 